Question 1: What is the place value of 5 in 3254710?

• A: None
• B: Only 1
• C: 1 and 2
• D: 2 and 3

Correct Answer: (c)

Solution: The place value of 5 in 3254710 is 50000. Hence, the correct answer is (c).

Question 2: The face value of 8 in the number

• A: 8
• B: 1000
• C: 8000
• D: 8926

Correct Answer: (a)

Solution: The face value of a digit is the digit itself. Hence, the face value of 8 is 8. The correct answer is (a).

Question 3: The sum of the place values of 3 in the number 503535 is

• A: 6
• B: 60
• C: 3030
• D: 3300

Correct Answer: (c)

Solution: Place values of 3 are 3000 and 30. Sum = 3000 + 30 = 3030. The correct answer is (c).

Question 4: The difference between the place values of 7 and 3 in 527435 is

• A: 4
• B: 45
• C: 64851
• D: 75142
• E: None of these

Correct Answer: (c)

Solution: Place value of 7 = 7000, Place value of 3 = 30. Difference = 7000 - 30 = 6970. The correct answer is (c).

Question 5: The difference between the local value and the face value of 7 in 32675149 is

• A: 5149
• B: 64851
• C: 69993
• D: 75142
• E: None of these

Correct Answer: (c)

Solution: Local value of 7 = 70000, Face value of 7 = 7. Difference = 70000 - 7 = 69993. The correct answer is (c).

Question 6: The sum of the greatest and smallest number of five digits is

• A: 11110
• B: 10999
• C: 109999
• D: 111110

Correct Answer: (d)

Solution: Greatest 5-digit number = 99999, Smallest 5-digit number = 10000. Sum = 99999 + 10000 = 109,999. The correct answer is (d).

Question 7: If the largest three-digit number is subtracted from the smallest five-digit number, then the remainder is

• A: 1
• B: 9000
• C: 9001
• D: 90001

Correct Answer: (c)

Solution: Smallest 5-digit number = 10000, Largest 3-digit number = 999. Remainder = 10000 - 999 = 9001. The correct answer is (c).

Question 8: The smallest number of 5 digits beginning with 3 and ending with 5 will be

• A: 31005
• B: 30015
• C: 30005
• D: 30025

Correct Answer: (c)

Solution: The smallest 5-digit number starting with 3 and ending with 5 is 30005. The correct answer is (c).

Question 9: What is the minimum number of four digits formed by using the digits 2, 4, 0, 7?

• A: 2047
• B: 2247
• C: 2407
• D: 2470

Correct Answer: (a)

Solution: The smallest 4-digit number is formed by arranging the digits in ascending order, ensuring the first digit is not zero. Thus, the smallest number is 2047. The correct answer is (a).

Question 10: All natural numbers and 0 are called the numbers.

• A: rational
• B: integer
• C: whole
• D: prime

Correct Answer: (c)

Solution: Natural numbers along with 0 are called whole numbers. The correct answer is (c).

Question 11: Consider the following statements about natural numbers:

• A: Only 1
• B: Only 2
• C: 1 and 2
• D: 2 and 3
• E: None of these

Correct Answer: (b)

Solution: Statement 1 is true (smallest natural number exists), but statement 2 is false (largest natural number does not exist). Statement 3 is also false. Hence, only statement 1 is correct. The correct answer is (b).

Question 12: Every rational number is also

• A: an integer
• B: a real number
• C: a natural number
• D: a whole number

Correct Answer: (b)

Solution: Every rational number is a real number, but it may not be an integer, natural number, or whole number. The correct answer is (b).

Question 13: The number n is

• A: a fraction
• B: a recurring decimal
• C: a rational number
• D: a real number

Correct Answer: (d)

Solution: The number n can be any real number. The correct answer is (d).

Question 14: √2 is a/an

• A: a finite decimal
• B: an infinite recurring decimal
• C: an irrational number
• D: a rational number

Correct Answer: (c)

Solution: √2 is an irrational number because it cannot be expressed as a fraction. The correct answer is (c).

Question 15: The number √3 is

• A: a finite decimal
• B: an infinite recurring decimal
• C: equal to 1.732
• D: an infinite non-recurring decimal

Correct Answer: (d)

Solution: √3 is an irrational number, which means it is an infinite non-recurring decimal. The correct answer is (d).

Question 16: In how many ways can 9 be expressed as the sum of two different positive integers?

• A: 3
• B: 4
• C: 5
• D: 6

Correct Answer: (b)

Solution: The pairs are (1,8), (2,7), (3,6), and (4,5). There are 4 ways. The correct answer is (b).

Question 17: P and Q are two positive integers such that PQ = 64. Which of the following cannot be the value of P + Q?

• A: 16
• B: 35
• C: 20
• D: 65

Correct Answer: (c)

Solution: Possible pairs (P, Q) are (1,64), (2,32), (4,16), (8,8). Their sums are 65, 34, 20, and 16. The sum 35 is not possible. The correct answer is (c).

Question 18: If x + y + z = 9 and both y and z are positive integers greater than zero, then the maximum value x can take is

• A: 3
• B: 7
• C: 8
• D: Data insufficient

Correct Answer: (c)

Solution: Since y and z are positive integers > 0, their minimum values are 1 each. Thus, x = 9 - (y + z) ≤ 9 - (1 + 1) = 7. The maximum value of x is 7. The correct answer is (c).

Question 19: What is the sum of the squares of the digits from 1 to 9?

• A: 105
• B: 260
• C: 285
• D: 385

Correct Answer: (d)

Solution: Sum = 1² + 2² + 3² + ... + 9² = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 = 385. The correct answer is (d).

Question 20: If n is an integer between 20 and 80, then any of the following could be n + 7 except

• A: 47
• B: 58
• C: 84
• D: 88

Correct Answer: (d)

Solution: n + 7 must lie between 27 and 87. 88 is outside this range. The correct answer is (d).

Question 21: Which one of the following is the correct sequence in respect of the Roman numerals: C, D, L and M?

• A: C > D > L > M
• B: M > L > D > C
• C: M > D > C > L
• D: L > C > D > M

Correct Answer: (c)

Solution: Roman numeral values: M = 1000, D = 500, C = 100, L = 50. The correct order is M > D > C > L. The correct answer is (c).

Question 22: If the numbers from 1 to 24, which are divisible by 2 are arranged in descending order, which number will be at the 8th place from the bottom?

• A: 10
• B: 12
• C: 16
• D: 18

Correct Answer: (c)

Solution: Numbers divisible by 2: 24, 22, 20, ..., 2. The 8th number from the bottom is 16. The correct answer is (c).

Question 23: 2 - 2 + 2 - 2 + ... (101 terms) = ?

• A: -2
• B: 0
• C: 2
• D: None of these

Correct Answer: (b)

Solution: The series alternates between 2 and -2. For 101 terms, there are 50 pairs of (2 - 2) = 0, plus one extra 2. Total = 0 + 2 = 2. The correct answer is (b).

Question 24: 98th term of the infinite series 1, 2, 3, 4, 1, 2, 3, 4, ... is

• A: 1
• B: 2
• C: 3
• D: 4

Correct Answer: (b)

Solution: The series repeats every 4 terms. 98 ÷ 4 leaves a remainder of 2, so the 98th term is the same as the 2nd term, which is 2. The correct answer is (b).

Question 25: If x, y, z are the digits of a number beginning from the left, the number is

• A: xyz
• B: x + 10y + 100z
• C: 10x + y + 100z
• D: 100x + 10y + z

Correct Answer: (d)

Solution: The positional values of the digits are: x (hundreds), y (tens), z (units). The number is 100x + 10y + z. The correct answer is (d).

Question 26: If x, y, z, w are the digits of a number beginning from the left, the number is

• A: xyzw
• B: wzyx
• C: x + 10y + 100z + 1000w
• D: 10³x + 10²y + 10z + w

Correct Answer: (d)

Solution: The positional values of the digits are: x (thousands), y (hundreds), z (tens), w (units). The number is 10³x + 10²y + 10z + w. The correct answer is (d).

Question 27: If n and p are both odd numbers, which of the following is an even number?

• A: n + p
• B: n + p + 1
• C: np + 2
• D: np

Correct Answer: (a)

Solution: The sum of two odd numbers (n + p) is always even. The correct answer is (a).

Question 28: For the integer n, if n³ is odd, then which of the following statements are true? I. n is odd. II. n² is odd. III. n² is even.

• A: I only
• B: II only
• C: I and II only
• D: I and III only

Correct Answer: (c)

Solution: If n³ is odd, n must be odd (I is true). The square of an odd number is odd (II is true). III is false. The correct answer is (c).

Question 29: If (n - 1) is an odd number, what are the two other odd numbers nearest to it?

• A: n, n - 1
• B: n, n - 2
• C: n - 3, n + 1
• D: n - 3, n + 5

Correct Answer: (c)

Solution: If (n - 1) is odd, n is even. The nearest odd numbers are n - 3 and n + 1. The correct answer is (c).

Question 30: Which of the following is always odd?

• A: Sum of two odd numbers
• B: Difference of two odd numbers
• C: Product of two odd numbers
• D: None of these

Correct Answer: (c)

Solution: The product of two odd numbers is always odd. The correct answer is (c).

Question 31: If x is an odd integer, then which of the following is true?

• A: 5x - 2 is even
• B: 5x² + 2 is odd
• C: 5x² + 3 is odd
• D: None of these

Correct Answer: (c)

Solution: For any odd integer x, 5x² is odd (odd × odd = odd), and adding 3 to it makes it even + odd = odd. Hence, 5x² + 3 is odd. The correct answer is (c).

Question 32: If A and B are two numbers such that AB = 0, then

• A: A = 0 and B = 0
• B: A = 0 or B = 0
• C: A = 0 and B ≠ 0
• D: B = 0 and A ≠ 0

Correct Answer: (b)

Solution: If AB = 0, either A = 0 or B = 0 or both. The correct answer is (b).

Question 33: If A, B, C, D are numbers in increasing order and D, B, E are in decreasing order, which sequence need neither be in decreasing nor increasing order?

• A: E, C, D
• B: E, B, C
• C: D, B, A
• D: A, E, C

Correct Answer: (d)

Solution: Analyzing the sequences, A, E, C does not follow a strict increasing or decreasing order. The correct answer is (d).

Question 34: If m, n, o, p, and q are integers, then m(n + o)(p - q) must be even when which of the following is even?

• A: m
• B: p
• C: m + n
• D: n + p

Correct Answer: (a)

Solution: For the product to be even, at least one factor must be even. Here, if m is even, the entire expression becomes even. The correct answer is (a).

Question 35: If n is a negative number, then which of the following is the least?

• A: 0
• B: -n
• C: 2n
• D: n²

Correct Answer: (c)

Solution: For a negative number n, -n is positive, 2n is more negative, and n² is always positive. Among the options, 2n is the least value. The correct answer is (c).

Question 36: If x - y = 8, then which of the following must be true?

• A: Both x and y are positive
• B: If x is positive, y must be positive
• C: If x is negative, y must be negative
• D: None of these

Correct Answer: (d)

Solution: If x - y = 8, x can be positive or negative, but y must adjust accordingly. Only statement III is always true. The correct answer is (d).

Question 37: If x and y are negative, then which of the following statements is/are always true? I. x + y is positive. II. xy is positive. III. x - y is positive.

• A: I only
• B: II only
• C: III only
• D: I and III only

Correct Answer: (b)

Solution: For negative x and y, x + y is negative, xy is positive, and x - y depends on values. Only II is always true. The correct answer is (b).

Question 38: If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true? I. n is odd. II. n is a perfect square. III. n is prime.

• A: I only
• B: I and II only
• C: I and III only
• D: None of these

Correct Answer: (c)

Solution: The product of four consecutive integers is always even, so n = 1 + even = odd. Also, n is a perfect square. The correct answer is (c).

Question 39: If x = -y + 3, how does y change when x increases from 1 to 2?

• A: y increases from -5 to -2.5
• B: y increases from 2.5 to 5
• C: y increases from 2.5 to 5
• D: y decreases from -5 to -2.5

Correct Answer: (a)

Solution: Solving for y when x = 1 and x = 2, we find y increases from -5 to -2.5. The correct answer is (a).

Question 40: If x is a rational number and y is an irrational number, then

• A: Both x + y and xy are necessarily rational
• B: Both x + y and xy are necessarily irrational
• C: xy is necessarily irrational, but x + y can be either rational or irrational
• D: x + y is necessarily irrational, but xy can be either rational or irrational

Correct Answer: (d)

Solution: Adding a rational and irrational number results in an irrational number, but multiplying may result in either. The correct answer is (d).

Question 41: The difference between the square of any two consecutive integers is equal to

• A: Sum of two numbers
• B: Difference of two numbers
• C: An even number
• D: Product of two numbers

Correct Answer: (a)

Solution: Let the integers be n and n+1. The difference is (n+1)² - n² = 2n + 1, which equals the sum of the two numbers. The correct answer is (a).

Question 42: Between two distinct rational numbers a and b, there exists another rational number which is

• A: a - b
• B: a + b
• C: (a + b)/2
• D: ab

Correct Answer: (c)

Solution: The average of two rational numbers is also rational. The correct answer is (c).

Question 43: If B > A, then which expression will have the highest value (given that A and B are positive integers)?

• A: A + B
• B: AB
• C: A² + B²
• D: Cannot say

Correct Answer: (d)

Solution: Without specific values of A and B, we cannot determine which expression is largest. The correct answer is (d).

Question 44: If 0 < x < 1, which of the following is greatest?

• A: x
• B: x²
• C: √x
• D: 1/x

Correct Answer: (d)

Solution: For 0 < x < 1, 1/x > √x > x > x². The correct answer is (d).

Question 45: If p is a positive fraction less than 1, then

• A: p² is less than 1
• B: p² is less than p
• C: p² is greater than p
• D: p² is equal to p

Correct Answer: (b)

Solution: For 0 < p < 1, p² < p. The correct answer is (b).

Question 46: If x is a real number, then x² + x + 1 is

• A: Always positive
• B: Zero for at least one value of x
• C: Always negative
• D: Greater than or equal to -4/3

Correct Answer: (a)

Solution: The discriminant of x² + x + 1 is negative, so it has no real roots and is always positive. The correct answer is (a).

Question 47: Let n be a natural number such that 1/n + 1/(n+1) + 1/(n+2) + ... + 1/(2n) is also a natural number. Which of the following statements is not true?

• A: 2 divides n
• B: 3 divides n
• C: 7 divides n
• D: n > 84

Correct Answer: (c)

Solution: Testing divisibility conditions, 7 does not divide n. The correct answer is (c).

Question 48: If n is an integer, how many values of n will give an integral value of (n² + 3n + 2)/6?

• A: 2
• B: 3
• C: 4
• D: None of these

Correct Answer: (b)

Solution: Factoring the numerator gives (n+1)(n+2)/6. For this to be an integer, n+1 or n+2 must be divisible by 6. There are 3 such values. The correct answer is (b).

Question 49: If p > q and r < 0, then which is true?

• A: pr < qr
• B: p - r < q - r
• C: p + r < q + r
• D: None of these

Correct Answer: (a)

Solution: Multiplying by a negative number reverses inequalities. Thus, pr < qr. The correct answer is (a).

Question 50: If X < Z and X < Y, which of the following is necessarily true?

• A: Y < Z
• B: ZX < Y + Z
• C: X² < YZ
• D: None of these

Correct Answer: (d)

Solution: From the given conditions, none of the options must be true. The correct answer is (d).

Question 51: In the relation x > y + z, x + y > p, and z < p, which of the following is necessarily true?

• A: y > p
• B: x + y > z
• C: y + p > x
• D: Insufficient data

Correct Answer: (d)

Solution: Without additional information, no conclusion can be drawn. The correct answer is (d).

Question 52: If a and b are positive integers and a/b = y, then

• A: b > a
• B: b < a
• C: b = a
• D: b/a = y

Correct Answer: (a)

Solution: If a/b = y and y > 1, then b > a. The correct answer is (a).

Question 53: If 13 = (1 - w)/w, then (2w)³ = ?

• A: -1
• B: 1
• C: 2
• D: -2

Correct Answer: (a)

Solution: Solving for w gives w = 1/14. Substituting into (2w)³ yields -1. The correct answer is (a).

Question 54: The second digit of the number is

• A: 5
• B: 7
• C: 9
• D: Cannot be determined

Correct Answer: (d)

Solution: Without knowing the full number, the second digit cannot be determined. The correct answer is (d).

Question 55: The last digit of the number is

• A: 0
• B: 1
• C: 2
• D: 3

Correct Answer: (c)

Solution: Based on the constraints, the last digit is 2. The correct answer is (c).

Question 56: The largest digit in the number is

• A: 5
• B: 7
• C: 8
• D: 9

Correct Answer: (d)

Solution: The largest digit is 9. The correct answer is (d).

Question 57: Which of the following is a factor of the given number?

• A: 2
• B: 3
• C: 4
• D: 9

Correct Answer: (a)

Solution: The number is divisible by 2. The correct answer is (a).

Question 58: The least prime number is

• A: 1
• B: 2
• C: 3
• D: 5

Correct Answer: (b)

Solution: The least prime number is 2. The correct answer is (b).

Question 59: Consider the following statements: 1. If x and y are composite numbers, then x + y is always composite. 2. There does not exist a natural number which is neither prime nor composite. Which of the above statements is/are correct?

• A: 1 only
• B: 2 only
• C: Both 1 and 2
• D: Neither 1 nor 2

Correct Answer: (b)

Solution: Statement 1 is false (e.g., 4 + 9 = 13, which is prime). Statement 2 is true. The correct answer is (b).

Question 60: The number of prime numbers between 0 and 50 is

• A: 14
• B: 15
• C: 16
• D: 17

Correct Answer: (b)

Solution: There are 15 prime numbers between 0 and 50. The correct answer is (b).

Question 61: The prime numbers dividing 143 and leaving a remainder of 3 in each case are

• A: 2 and 11
• B: 11 and 13
• C: 3 and 7
• D: 5 and 7

Correct Answer: (b)

Solution: Dividing 143 by 11 and 13 leaves a remainder of 3 in each case. The correct answer is (b).

Question 62: The sum of the first four primes is

• A: 10
• B: 11
• C: 16
• D: 17

Correct Answer: (a)

Solution: The first four primes are 2, 3, 5, and 7. Their sum is 17. The correct answer is (a).

Question 63: The sum of all the prime numbers from 1 to 20 is

• A: 75
• B: 76
• C: 77
• D: 78

Correct Answer: (c)

Solution: The primes from 1 to 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Their sum is 77. The correct answer is (c).

Question 64: A prime number N, in the range 10 to 50, remains unchanged when its digits are reversed. The square of such a number is

• A: 121
• B: 484
• C: 1089
• D: 1936

Correct Answer: (b)

Solution: The only such prime is 11. Its square is 121. The correct answer is (b).

Question 65: The remainder obtained when any prime number greater than 6 is divided by 6 must be

• A: Either 1 or 2
• B: Either 1 or 3
• C: Either 1 or 5
• D: Either 3 or 5

Correct Answer: (c)

Solution: Any prime greater than 6 is of the form 6k ± 1. The remainder is either 1 or 5. The correct answer is (c).

Question 66: Which of the following is not a prime number?

• A: 21
• B: 23
• C: 29
• D: 43

Correct Answer: (a)

Solution: 21 is not a prime number (divisible by 3). The correct answer is (a).

Question 67: Which of the following is a prime number?

• A: 19
• B: 20
• C: 21
• D: 22

Correct Answer: (a)

Solution: 19 is a prime number. The correct answer is (a).

Question 68: Which of the following is a prime number?

• A: 115
• B: 119
• C: 127
• D: None of these

Correct Answer: (c)

Solution: 127 is a prime number because it has no divisors other than 1 and itself. The correct answer is (c).

Question 69: Which of the following is a prime number?

• A: 289
• B: 359
• C: 449
• D: None of these

Correct Answer: (b)

Solution: 359 is a prime number. The correct answer is (b).

Question 70: Which of the following is a prime number?

• A: 143
• B: 117
• C: 127
• D: None of these

Correct Answer: (c)

Solution: 127 is a prime number. The correct answer is (c).

Question 71: The smallest value of natural number n, for which 2n + 1 is not a prime number, is

• A: 3
• B: 4
• C: 5
• D: 6

Correct Answer: (a)

Solution: For n = 3, 2n + 1 = 7 (prime). For n = 4, 2n + 1 = 9 (not prime). The smallest value of n is 4. The correct answer is (a).

Question 72: The smallest three-digit prime number is

• A: 101
• B: 103
• C: 107
• D: None of these

Correct Answer: (a)

Solution: The smallest three-digit number is 100, but it is not prime. The next number, 101, is prime. The correct answer is (a).

Question 73: How many of the integers between 110 and 120 are prime numbers?

• A: 0
• B: 1
• C: 2
• D: 3

Correct Answer: (c)

Solution: Checking each number: 111, 112, 114, 115, 116, 117, 118, and 119 are not prime. Only 113 and 127 are prime. The correct answer is (c).

Question 74: Four prime numbers are arranged in ascending order. The product of the first three is 385 and that of the last three is 1001. The largest prime number is

• A: 9
• B: 11
• C: 13
• D: 17

Correct Answer: (b)

Solution: Let the primes be p, q, r, s. From pqrs = 385 and qrs = 1001, we find s = 11. The correct answer is (b).

Question 75: Which one of the following is a prime number?

• A: 161
• B: 221
• C: 373
• D: 437

Correct Answer: (c)

Solution: 161 = 7 × 23, 221 = 13 × 17, 373 is prime, 437 = 19 × 23. The correct answer is (c).

Question 76: The smallest prime number, that is the fifth term of an increasing arithmetic sequence in which all the four preceding terms are also prime, is

• A: 17
• B: 29
• C: 37
• D: 53

Correct Answer: (a)

Solution: The sequence is 5, 11, 17, 23, 29. The fifth term is 29. The correct answer is (a).

Question 77: The number of prime numbers between 301 and 320 are

• A: 3
• B: 4
• C: 5
• D: 6

Correct Answer: (b)

Solution: The primes are 307, 311, 313, 317. There are 4 primes. The correct answer is (b).

Question 78: Consider the following statements: 1. If p > 2 is a prime, then it can be written as 4n + 1 or 4n + 3 for a suitable natural number n. 2. If p > 2 is a prime, then (p - 1)(p + 1) is always divisible by 4. Of these statements,

• A: (1) is true but (2) is false
• B: (1) is false but (2) is true
• C: (1) and (2) are false
• D: (1) and (2) are true

Correct Answer: (d)

Solution: Both statements are true. Any odd prime can be expressed as 4n + 1 or 4n + 3, and (p - 1)(p + 1) is divisible by 4 for any odd p. The correct answer is (d).

Question 79: What is the first value of n for which n² + n + 41 is not a prime?

• A: 1
• B: 10
• C: 20
• D: 40

Correct Answer: (d)

Solution: For n = 40, n² + n + 41 = 1681, which is not prime (41 × 41). The correct answer is (d).

Question 80: Let Xₖ = {p₁p₂...pₖ} + 1, where p₁, p₂, ..., pₖ are the first k primes. Consider the following: 1. Xₖ is a prime number. 2. Xₖ is a composite number. 3. Xₖ + 1 is always an even number. Which of the above is/are correct?

• A: 1 only
• B: 2 only
• C: 3 only
• D: 1 and 3

Correct Answer: (c)

Solution: Xₖ is not necessarily prime (e.g., X₅ = 2311 is prime, but X₆ = 30031 is composite). Xₖ + 1 is always even. The correct answer is (c).

Question 81: 6 × 3(3 - 1) is equal to

• A: 19
• B: 20
• C: 36
• D: 53

Correct Answer: (c)

Solution: Simplifying: 6 × 3(3 - 1) = 6 × 3 × 2 = 36. The correct answer is (c).

Question 82: What is 394 times 113?

• A: 44402
• B: 44522
• C: 44632
• D: None of these

Correct Answer: (b)

Solution: 394 × 113 = 44522. The correct answer is (b).

Question 83: 14 ÷ 9 = ?

• A: 9
• B: 10
• C: 81
• D: 810

Correct Answer: (a)

Solution: 14 ÷ 9 = 1 remainder 5. The correct answer is (a).

Question 84: 136 × 12 × 8 = ?

• A: 12066
• B: 13046
• C: 13064
• D: 13066

Correct Answer: (d)

Solution: 136 × 12 × 8 = 13066. The correct answer is (d).

Question 85: 8888 + 848 + 88 - ? = 7337 + 737

• A: 1450
• B: 1550
• C: 1650
• D: 1750

Correct Answer: (c)

Solution: Solving: 8888 + 848 + 88 - x = 7337 + 737 → x = 1650. The correct answer is (c).

Question 86: 414 × ? × 7 = 127512

• A: 36
• B: 40
• C: 44
• D: 48

Correct Answer: (c)

Solution: Solving: 414 × x × 7 = 127512 → x = 44. The correct answer is (c).

Question 87: Product of 82540027 and 43253 is

• A: 3570103787831
• B: 3570103787832
• C: 3570103787833
• D: 3570103787834

Correct Answer: (a)

Solution: Direct multiplication gives 3570103787831. The correct answer is (a).

Question 88: (46351 - 36418 - 4505) ÷ ? = 1357

• A: 2
• B: 3
• C: 4
• D: 6

Correct Answer: (a)

Solution: Solving: (46351 - 36418 - 4505) ÷ x = 1357 → x = 2. The correct answer is (a).

Question 89: The value of 112 × 54 is

• A: 6700
• B: 70000
• C: 76500
• D: 77200

Correct Answer: (b)

Solution: 112 × 54 = 6048. The correct answer is (b).

Question 90: Multiply 5746320819 by 125

• A: 718290102375
• B: 728490301375
• C: 748290103375
• D: 798290102975

Correct Answer: (a)

Solution: 5746320819 × 125 = 718290102375. The correct answer is (a).

Question 91: 935421 × 625 = ?

• A: 575648125
• B: 584638125
• C: 584649125
• D: 585628125

Correct Answer: (c)

Solution: 935421 × 625 = 584649125. The correct answer is (c).

Question 92: (999)² - (998)² = ?

• A: 1992
• B: 1995
• C: 1997
• D: 1998

Correct Answer: (b)

Solution: Using the difference of squares: (999)² - (998)² = (999 - 998)(999 + 998) = 1997. The correct answer is (b).

Question 93: (80)² - (65)² + 81 = ?

• A: 2094
• B: 2256
• C: 306
• D: 2175

Correct Answer: (a)

Solution: Simplifying: (80)² - (65)² + 81 = 6400 - 4225 + 81 = 2094. The correct answer is (a).

Question 94: (65)² - (55)² = ?

• A: 10
• B: 100
• C: 120
• D: 1200

Correct Answer: (b)

Solution: Using the difference of squares: (65)² - (55)² = (65 - 55)(65 + 55) = 10 × 120 = 1200. The correct answer is (b).

Question 95: If a and b are positive integers such that a² - b² = 19, then the value of a is

• A: 9
• B: 10
• C: 19
• D: 20

Correct Answer: (b)

Solution: Factoring: a² - b² = (a - b)(a + b) = 19. Since 19 is prime, a - b = 1 and a + b = 19 → a = 10. The correct answer is (b).

Question 96: If a and b are positive integers, a > b and (a + b)² - (a - b)² > 29, then the smallest value of a is

• A: 3
• B: 4
• C: 6
• D: 7

Correct Answer: (d)

Solution: Expanding: (a + b)² - (a - b)² = 4ab > 29 → ab > 7.25. The smallest integer values are a = 7, b = 1. The correct answer is (d).

Question 97: 397 × 397 + 104 × 104 + 2 × 397 × 104 = ?

• A: 250001
• B: 251001
• C: 260101
• D: 261001

Correct Answer: (b)

Solution: This is a perfect square expansion: (397 + 104)² = 501² = 251001. The correct answer is (b).

Question 98: If (64)² - (36)² = 20 × x, then x = ?

• A: 70
• B: 120
• C: 180
• D: None of these

Correct Answer: (a)

Solution: Simplifying: (64)² - (36)² = (64 - 36)(64 + 36) = 28 × 100 = 20 × x → x = 70. The correct answer is (a).

Question 99: (489 + 375)² - (489 - 375)² ÷ (489 × 375) = ?

• A: 864
• B: 4
• C: None of these

Correct Answer: (a)

Solution: Using the identity: (a + b)² - (a - b)² = 4ab → 4 × 489 × 375 ÷ (489 × 375) = 4. The correct answer is (a).

Question 100: -95 ÷ 19 = ?

• A: -5
• B: 0
• C: 5
• D: None of these

Correct Answer: (a)

Solution: Dividing -95 by 19 gives -5. The correct answer is (a).

Question 101: 12345679 × 72 is equal to

• A: 88888888
• B: 888888888
• C: 898989898
• D: 999999998

Correct Answer: (b)

Solution: Multiplying 12345679 by 72 gives 888888888. The correct answer is (b).

Question 102: 8899 - 6644 - 3322 = ? - 1122

• A: -43
• B: -48
• C: -17
• D: -20

Correct Answer: (a)

Solution: Simplifying: 8899 - 6644 - 3322 = -1067. Adding 1122 gives -43. The correct answer is (a).

Question 103: 74844 ÷ ? = 54 × 63

• A: 22
• B: 34
• C: 42
• D: None of these

Correct Answer: (a)

Solution: Solving: 74844 ÷ x = 54 × 63 → x = 22. The correct answer is (a).

Question 104: 1256 × 3892 = ?

• A: 4883852
• B: 4888532
• C: 4888352
• D: 4883582

Correct Answer: (c)

Solution: Direct multiplication gives 4888352. The correct answer is (c).

Question 105: What is 786 times 964?

• A: 757704
• B: 754164
• C: 749844
• D: 749844

Correct Answer: (a)

Solution: Multiplying 786 by 964 gives 757704. The correct answer is (a).

Question 106: What is 348 times 265?

• A: 88740
• B: 89750
• C: 92220
• D: 95700

Correct Answer: (c)

Solution: Multiplying 348 by 265 gives 92220. The correct answer is (c).

Question 107: (71 × 29 + 27 × 15 + 8 × 4) equals

• A: 2496
• B: 3450
• C: 3458
• D: None of these

Correct Answer: (c)

Solution: Simplifying: (71 × 29) + (27 × 15) + (8 × 4) = 2059 + 405 + 32 = 3458. The correct answer is (c).

Question 108: 106 × 106 - 94 × 94 = ?

• A: 2400
• B: 273268
• C: 2029272
• D: 1923472

Correct Answer: (a)

Solution: Using the difference of squares: (106² - 94²) = (106 - 94)(106 + 94) = 12 × 200 = 2400. The correct answer is (a).

Question 109: 8796 × 223 + 8796 × 77 = ?

• A: 273268
• B: 273358
• C: 2738303
• D: 2731703

Correct Answer: (a)

Solution: Factoring: 8796 × (223 + 77) = 8796 × 300 = 273268. The correct answer is (a).

Question 110: 287 × 287 + 269 × 269 - 2 × 287 × 269 = ?

• A: 2736900
• B: 2738800
• C: 2716740
• D: None of these

Correct Answer: (c)

Solution: This is a perfect square expansion: (287 - 269)² = 18² = 324. The correct answer is (c).

Question 111: {(476 + 424)² - 4 × 476 × 424} = ?

• A: 2906
• B: 3116
• C: 2704
• D: None of these

Correct Answer: (c)

Solution: Simplifying: (476 + 424)² - 4 × 476 × 424 = (900)² - 4 × 476 × 424 = 810000 - 808256 = 1744. The correct answer is (c).

Question 112: The value of 112 × 54 is

• A: 6700
• B: 70000
• C: 76500
• D: 77200

Correct Answer: (b)

Solution: Multiplying: 112 × 54 = 6048. The correct answer is (b).

Question 113: Multiply 5746320819 by 125

• A: 718290102375
• B: 728490301375
• C: 748290103375
• D: 798290102975

Correct Answer: (a)

Solution: Multiplying: 5746320819 × 125 = 718290102375. The correct answer is (a).

Question 114: 935421 × 625 = ?

• A: 575648125
• B: 584638125
• C: 584649125
• D: 585628125

Correct Answer: (c)

Solution: Multiplying: 935421 × 625 = 584649125. The correct answer is (c).

Question 115: (999)² - (998)² = ?

• A: 1992
• B: 1995
• C: 1997
• D: 1998

Correct Answer: (c)

Solution: Using the difference of squares: (999)² - (998)² = (999 - 998)(999 + 998) = 1 × 1997 = 1997. The correct answer is (c).

Question 116: (80)² - (65)² + 81 = ?

• A: 306
• B: 2094
• C: 2256
• D: None of these

Correct Answer: (b)

Solution: Simplifying: (80)² - (65)² + 81 = 6400 - 4225 + 81 = 2094. The correct answer is (b).

Question 117: (65)² - (55)² = ?

• A: 10
• B: 100
• C: 120
• D: 1200

Correct Answer: (d)

Solution: Using the difference of squares: (65)² - (55)² = (65 - 55)(65 + 55) = 10 × 120 = 1200. The correct answer is (d).

Question 118: If a and b are positive integers such that a² - b² = 19, then the value of a is

• A: 9
• B: 10
• C: 19
• D: 20

Correct Answer: (b)

Solution: Factoring: a² - b² = (a - b)(a + b) = 19. Since 19 is prime, a - b = 1 and a + b = 19 → a = 10. The correct answer is (b).

Question 119: If a and b are positive integers, a > b and (a + b)² - (a - b)² > 29, then the smallest value of a is

• A: 3
• B: 4
• C: 6
• D: 7

Correct Answer: (d)

Solution: Expanding: (a + b)² - (a - b)² = 4ab > 29 → ab > 7.25. The smallest integer values are a = 7, b = 1. The correct answer is (d).

Question 120: 397 × 397 + 104 × 104 + 2 × 397 × 104 = ?

• A: 250001
• B: 251001
• C: 260101
• D: 261001

Correct Answer: (b)

Solution: This is a perfect square expansion: (397 + 104)² = 501² = 251001. The correct answer is (b).

Question 121: If (64)² - (36)² = 20 × x, then x = ?

• A: 70
• B: 120
• C: 180
• D: None of these

Correct Answer: (a)

Solution: Simplifying: (64)² - (36)² = (64 - 36)(64 + 36) = 28 × 100 = 20 × x → x = 70. The correct answer is (a).

Question 122: (489 + 375)² - (489 - 375)² ÷ (489 × 375) = ?

• A: 864
• B: 4
• C: None of these

Correct Answer: (a)

Solution: Using the identity: (a + b)² - (a - b)² = 4ab → 4 × 489 × 375 ÷ (489 × 375) = 4. The correct answer is (a).

Question 123: From the sum of 17 and -12, subtract 48.

• A: -43
• B: -48
• C: -17
• D: -20

Correct Answer: (a)

Solution: Simplifying: (17 - 12) - 48 = 5 - 48 = -43. The correct answer is (a).

Question 124: 60840 ÷ 234 = ?

• A: 225
• B: 255
• C: 260
• D: 310

Correct Answer: (c)

Solution: Dividing: 60840 ÷ 234 = 260. The correct answer is (c).

Question 125: (24 + 25 + 26)² - (10 + 20 + 25)² = ?

• A: 352
• B: 400
• C: 2600
• D: None of these

Correct Answer: (c)

Solution: Simplifying: (75)² - (55)² = (75 - 55)(75 + 55) = 20 × 130 = 2600. The correct answer is (c).

Question 126: If a = 11 and b = 5, then (a² + b² + ab) ÷ (a³ - b³) = ?

• A: 2
• B: 5
• C: 10
• D: 20

Correct Answer: (a)

Solution: Substituting: (11² + 5² + 11 × 5) ÷ (11³ - 5³) = (121 + 25 + 55) ÷ (1331 - 125) = 201 ÷ 1206 = 2. The correct answer is (a).

Question 127: If a + b + c = 0, then (a + b)(b + c)(c + a) equals

• A: ab(a + b)
• B: (a + b + c)²
• C: -abc
• D: a² + b² + c²

Correct Answer: (c)

Solution: If a + b + c = 0, then (a + b)(b + c)(c + a) = -abc. The correct answer is (c).

Question 128: If a = 7, b = 5, c = 3, then the value of a² + b² + c² - ab - bc - ca is

• A: -12
• B: 8
• C: 12
• D: 20

Correct Answer: (b)

Solution: Substituting: 7² + 5² + 3² - (7 × 5) - (5 × 3) - (3 × 7) = 49 + 25 + 9 - 35 - 15 - 21 = 8. The correct answer is (b).

Question 129: Both addition and multiplication of numbers are operations which are

• A: Neither commutative nor associative
• B: Associative but not commutative
• C: Commutative but not associative
• D: Commutative and associative

Correct Answer: (d)

Solution: Addition and multiplication are both commutative and associative. The correct answer is (d).

Question 130: Which of the following digits will replace the H marks in the equation: 9H + H8 + H6 = 230?

• A: 3
• B: 4
• C: 5
• D: 9

Correct Answer: (c)

Solution: Testing: 95 + 58 + 56 = 230. The correct answer is (c).

Question 131: If a and b are positive integers such that a² - b² = 19, then the value of a is

• A: 9
• B: 10
• C: 19
• D: 20

Correct Answer: (b)

Solution: Factoring: a² - b² = (a - b)(a + b) = 19. Since 19 is prime, a - b = 1 and a + b = 19 → a = 10. The correct answer is (b).

Question 132: If a and b are positive integers, a > b and (a + b)² - (a - b)² > 29, then the smallest value of a is

• A: 3
• B: 4
• C: 6
• D: 7

Correct Answer: (d)

Solution: Expanding: (a + b)² - (a - b)² = 4ab > 29 → ab > 7.25. The smallest integer values are a = 7, b = 1. The correct answer is (d).

Question 133: 397 × 397 + 104 × 104 + 2 × 397 × 104 = ?

• A: 250001
• B: 251001
• C: 260101
• D: 261001

Correct Answer: (b)

Solution: This is a perfect square expansion: (397 + 104)² = 501² = 251001. The correct answer is (b).

Question 134: If (64)² - (36)² = 20 × x, then x = ?

• A: 70
• B: 120
• C: 180
• D: None of these

Correct Answer: (a)

Solution: Simplifying: (64)² - (36)² = (64 - 36)(64 + 36) = 28 × 100 = 20 × x → x = 70. The correct answer is (a).

Question 135: (489 + 375)² - (489 - 375)² ÷ (489 × 375) = ?

• A: 864
• B: 4
• C: None of these

Correct Answer: (a)

Solution: Using the identity: (a + b)² - (a - b)² = 4ab → 4 × 489 × 375 ÷ (489 × 375) = 4. The correct answer is (a).

Question 136: From the sum of 17 and -12, subtract 48.

• A: -43
• B: -48
• C: -17
• D: -20

Correct Answer: (a)

Solution: Simplifying: (17 - 12) - 48 = 5 - 48 = -43. The correct answer is (a).

Question 137: 60840 ÷ 234 = ?

• A: 225
• B: 255
• C: 260
• D: 310

Correct Answer: (c)

Solution: Dividing: 60840 ÷ 234 = 260. The correct answer is (c).

Question 138: (24 + 25 + 26)² - (10 + 20 + 25)² = ?

• A: 352
• B: 400
• C: 2600
• D: None of these

Correct Answer: (c)

Solution: Simplifying: (75)² - (55)² = (75 - 55)(75 + 55) = 20 × 130 = 2600. The correct answer is (c).

Question 139: (65)² - (55)² = ?

• A: 10
• B: 100
• C: 120
• D: 1200

Correct Answer: (d)

Solution: Using the difference of squares: (65)² - (55)² = (65 - 55)(65 + 55) = 10 × 120 = 1200. The correct answer is (d).

Question 140: Both addition and multiplication of numbers are operations which are

• A: Neither commutative nor associative
• B: Associative but not commutative
• C: Commutative but not associative
• D: Commutative and associative

Correct Answer: (d)

Solution: Addition and multiplication are both commutative and associative. The correct answer is (d).

Question 141: Which of the following digits will replace the H marks in the equation: 9H + H8 + H6 = 230?

• A: 3
• B: 4
• C: 5
• D: 9

Correct Answer: (c)

Solution: Testing: 95 + 58 + 56 = 230. The correct answer is (c).

Question 142: Find the missing number in the addition problem: 8 + 4 + 9 + 2 + 2 + 3 + * = 58.

• A: 0
• B: 4
• C: 6
• D: 9

Correct Answer: (b)

Solution: Adding: 8 + 4 + 9 + 2 + 2 + 3 = 28. To make it 58, the missing number is 30. The correct answer is (b).

Question 143: What is the minimum number of four digits formed by using the digits 2, 4, 0, 7?

• A: 2047
• B: 2247
• C: 2407
• D: 2470

Correct Answer: (a)

Solution: Arranging the digits in ascending order gives 2047. The correct answer is (a).

Question 144: All natural numbers and 0 are called the numbers.

• A: Rational
• B: Integer
• C: Whole
• D: Prime

Correct Answer: (c)

Solution: Natural numbers and 0 are called whole numbers. The correct answer is (c).

Question 145: Consider the following statements about natural numbers: (1) There exists a smallest natural number. (2) There exists a largest natural number. (3) Between two natural numbers, there is always a natural number. Which of the above statements is/are correct?

• A: (1) Only
• B: (2) Only
• C: (1) and (2) Only
• D: (3) Only

Correct Answer: (a)

Solution: Statement (1) is correct because 1 is the smallest natural number. Statements (2) and (3) are incorrect. The correct answer is (a).

Question 146: Every rational number is also

• A: An integer
• B: A real number
• C: A natural number
• D: A whole number

Correct Answer: (b)

Solution: Every rational number is a real number. The correct answer is (b).

Question 147: The number n is

• A: A fraction
• B: A rational number
• C: An irrational number
• D: A whole number

Correct Answer: (c)

Solution: √2 is an irrational number. The correct answer is (c).

Question 148: √3 is

• A: A finite decimal
• B: An infinite recurring decimal
• C: Equal to 1.732
• D: An infinite non-recurring decimal

Correct Answer: (d)

Solution: √3 is an infinite non-recurring decimal. The correct answer is (d).

Question 149: There are just two ways in which 5 may be expressed as the sum of two different positive integers. In how many ways can 9 be expressed as the sum of two different positive integers?

• A: 3
• B: 4
• C: 5
• D: 6

Correct Answer: (b)

Solution: The pairs are (1,8), (2,7), (3,6), (4,5). Total = 4. The correct answer is (b).

Question 150: P and Q are two positive integers such that PQ = 64. Which of the following cannot be the value of P + Q?

• A: 16
• B: 35
• C: 65
• D: 80

Correct Answer: (c)

Solution: Testing: P + Q = 16 (P=8, Q=8), 35 (P=32, Q=2), 65 (not possible), 80 (P=64, Q=1). The correct answer is (c).

Question 151: If x + y + z = 9 and both y and z are positive integers greater than zero, then the maximum value x can take is

• A: 3
• B: 7
• C: 8
• D: Data insufficient

Correct Answer: (c)

Solution: Maximizing x: y = 1, z = 1 → x = 9 - 1 - 1 = 7. The correct answer is (c).

Question 152: What is the sum of the squares of the digits from 1 to 9?

• A: 105
• B: 260
• C: 285
• D: 385

Correct Answer: (d)

Solution: Sum = 1² + 2² + ... + 9² = 385. The correct answer is (d).

Question 153: If n is an integer between 20 and 80, then any of the following could be n + 7 except

• A: 47
• B: 58
• C: 84
• D: 88

Correct Answer: (d)

Solution: Testing: n + 7 = 47 → n = 40, n + 7 = 58 → n = 51, n + 7 = 84 → n = 77, n + 7 = 88 → n = 81 (not in range). The correct answer is (d).

Question 154: Which one of the following is the correct sequence in respect of the Roman numerals: C, D, L and M?

• A: C > D > L > M
• B: M > L > D > C
• C: M > D > C > L
• D: L > C > D > M

Correct Answer: (c)

Solution: M = 1000, D = 500, C = 100, L = 50. The correct order is M > D > C > L. The correct answer is (c).

Question 155: If the numbers from 1 to 24, which are divisible by 2 are arranged in descending order, which number will be at the 8th place from the bottom?

• A: 10
• B: 12
• C: 16
• D: 18

Correct Answer: (c)

Solution: Numbers: 24, 22, 20, ..., 2. The 8th from the bottom is 16. The correct answer is (c).

Question 156: 2 - 2 + 2 - 2 + ... (101 terms) = ?

• A: -2
• B: 0
• C: 2
• D: None of these

Correct Answer: (c)

Solution: Alternating series: 2 - 2 + 2 - 2 + ... (101 terms) = 2. The correct answer is (c).

Question 157: If m, n, o, p and q are integers, then m(n + 6)(p - q) must be even when which of the following is even?

• A: m
• B: p
• C: m + n
• D: n + p

Correct Answer: (a)

Solution: For the product to be even, at least one factor must be even. If m is even, the product is even. The correct answer is (a).

Question 158: If n is a negative number, then which of the following is the least?

• A: 0
• B: -n
• C: 2n
• D: n²

Correct Answer: (c)

Solution: For n < 0, 2n is the least. The correct answer is (c).

Question 159: If x - y = 8, then which of the following must be true?

• A: Both x and y are positive
• B: If x is positive, y must be positive
• C: If x is negative, y must be negative
• D: None of these

Correct Answer: (c)

Solution: If x is negative, y = x - 8 is also negative. The correct answer is (c).

Question 160: If x and y are negative, then which of the following statements is/are always true? I. x + y is positive. II. xy is positive. III. x - y is positive.

• A: I only
• B: II only
• C: III only
• D: I and II only

Correct Answer: (b)

Solution: For x, y < 0, x + y < 0, xy > 0, x - y depends on values. Only II is always true. The correct answer is (b).

Question 161: If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true? I. n is odd. II. n is prime. III. n is a perfect square.

• A: I only
• B: I and II only
• C: I and III only
• D: None of these

Correct Answer: (c)

Solution: x is even → n = 1 + x is odd. x is divisible by 4 → n is not prime. x + 1 is a perfect square. The correct answer is (c).

Question 162: If x is a real number, then x² + x + 1 is

• A: Always negative
• B: Zero for at least one value of x
• C: Greater than or equal to 3/4
• D: None of these

Correct Answer: (c)

Solution: Completing the square: x² + x + 1 = (x + ½)² + ¾ ≥ ¾. The correct answer

Question 163: The number of prime factors in the expression (15)^12 × (7)^5 × (13)^3 is

• A: 20
• B: 25
• C: 30
• D: 35

Correct Answer: (c)

Solution: Total prime factors = 12 (from 15) + 5 (from 7) + 3 (from 13) = 30. The correct answer is (c).

Question 164: What number multiplied by 48 will give the same product as 173 × 240?

• A: 495
• B: 545
• C: 685
• D: 865

Correct Answer: (d)

Solution: Let x × 48 = 173 × 240 → x = (173 × 240) ÷ 48 = 865. The correct answer is (d).

Question 165: A positive number, which when added to 1000, gives a sum greater than when it is multiplied by 1000. This positive integer is

• A: 1
• B: 3
• C: 5
• D: 7

Correct Answer: (a)

Solution: Let the number be x. Then, 1000 + x > 1000x → x < 1. The only positive integer satisfying this is x = 1. The correct answer is (a).

Question 166: 7 is added to a certain number; the sum is multiplied by 5; the product is divided by 9, and 3 is subtracted from the quotient. If the remainder left is 12, what was the original number?

• A: 20
• B: 30
• C: 40
• D: 60

Correct Answer: (c)

Solution: Reverse the operations: (12 + 3) × 9 ÷ 5 - 7 = 40. The correct answer is (c).

Question 167: Symbiosis runs a Corporate Training Programme. At the end of running the first programme, its total takings were ₹38950. There were more than 45 but less than 100 participants. What was the participant fee for the programme?

• A: 410
• B: 450
• C: 500
• D: 510

Correct Answer: (a)

Solution: Let the fee be x and participants be n. Then, n × x = 38950. Testing values: n = 95 → x = 410. The correct answer is (a).

Question 168: The sum of four consecutive even numbers A, B, C, and D is 180. What is the sum of the set of next four consecutive even numbers?

• A: 196
• B: 204
• C: 212
• D: 214

Correct Answer: (b)

Solution: Let the numbers be x, x+2, x+4, x+6. Sum = 4x + 12 = 180 → x = 42. Next four numbers are 48, 50, 52, 54. Their sum = 204. The correct answer is (b).

Question 169: A young girl counted on her fingers. She started calling the thumb 1, index finger 2, middle finger 3, ring finger 4, little finger 5, then reversed direction. She counted up to 1994. On which finger did she end?

• A: Thumb
• B: Index finger
• C: Middle finger
• D: Ring finger

Correct Answer: (b)

Solution: The sequence repeats every 8 counts. 1994 ÷ 8 = 249 remainder 2. The second finger is the index finger. The correct answer is (b).

Question 170: Given n = 1 + x and x is the product of four consecutive integers. Which of the following is true? I. n is an odd integer. II. n is prime. III. n is a perfect square.

• A: Only I
• B: Only III
• C: Both I and II
• D: Both I and III

Correct Answer: (d)

Solution: x is divisible by 4 → n = 1 + x is odd. x is not prime → n is not prime. x + 1 is a perfect square. The correct answer is (d).

Question 171: If x + y = 15 and xy = 56, then what is the value of x² + y²?

• A: 110
• B: 113
• C: 121
• D: Cannot be determined

Correct Answer: (b)

Solution: Using the identity: x² + y² = (x + y)² - 2xy = 15² - 2(56) = 225 - 112 = 113. The correct answer is (b).

Question 172: Given that (1² + 2² + 3² + ... + 20²) = 2870, the value of (2² + 4² + 6² + ... + 40²) is

• A: 2870
• B: 5740
• C: 11480
• D: 28700

Correct Answer: (c)

Solution: Factor out 4: 4(1² + 2² + 3² + ... + 20²) = 4 × 2870 = 11480. The correct answer is (c).

Question 173: The value of 5² + 6² + ... + 10² + 20² is

• A: 755
• B: 760
• C: 765
• D: 770

Correct Answer: (c)

Solution: Compute directly: 25 + 36 + 49 + 64 + 81 + 100 + 400 = 765. The correct answer is (c).

Question 174: Given that 1 + 2 + 3 + ... + 10 = 55, the sum 6 + 12 + 18 + ... + 60 is equal to

• A: 300
• B: 330
• C: 455
• D: 655

Correct Answer: (b)

Solution: Factor out 6: 6(1 + 2 + 3 + ... + 10) = 6 × 55 = 330. The correct answer is (b).

Question 175: If m and n are natural numbers such that 2ᵐ - 2ⁿ = 960, what is the value of m?

• A: 10
• B: 12
• C: 15
• D: Cannot be determined

Correct Answer: (a)

Solution: Factorize: 2ⁿ(2ᵐ⁻ⁿ - 1) = 960. Testing values: n = 6, m = 10 → 2⁶(2⁴ - 1) = 960. The correct answer is (a).

Question 176: On multiplying a number by 7, all the digits in the product appear as 3's. The smallest such number is

• A: 47619
• B: 46719
• C: 48619
• D: 47649

Correct Answer: (a)

Solution: Let the number be x. Then, 7x = 333...3. Testing values: x = 47619 → 7 × 47619 = 333333. The correct answer is (a).

Question 177: The number of digits in the smallest number, which when multiplied by 7 yields all nines, is

• A: 3
• B: 4
• C: 5
• D: 6

Correct Answer: (c)

Solution: Let the number be x. Then, 7x = 999...9. Testing values: x = 142857 → 7 × 142857 = 999999. The correct answer is (c).

Question 178: A boy multiplies 987 by a certain number and obtains 559981 as his answer. If in the answer both 9's are wrong but the other digits are correct, then the correct answer will be

• A: 553681
• B: 555181
• C: 555681
• D: 556581

Correct Answer: (c)

Solution: Replace the incorrect digits: 559981 → 555681. The correct answer is (c).

Question 179: The numbers 1, 3, 5, ..., 25 are multiplied together. The number of zeros at the right end of the product is

• A: 0
• B: 1
• C: 2
• D: 3

Correct Answer: (a)

Solution: No factor of 10 exists in the product. The correct answer is (a).

Question 180: The numbers 2, 4, 6, ..., 100 are multiplied together. The number of zeros at the end of the product is

• A: 10
• B: 11
• C: 12
• D: 13

Correct Answer: (c)

Solution: Count factors of 5: 10, 20, 30, ..., 100 → 12 zeros. The correct answer is (c).

Question 181: Let S be the set of prime numbers ≥ 2 and < 100. Multiply all elements of S. With how many consecutive zeros will the product end?

• A: 1
• B: 2
• C: 5
• D: 10

Correct Answer: (a)

Solution: No factor of 10 exists in the product. The correct answer is (a).

Question 182: Find the number of zeros at the end of the result of 3 × 6 × 9 × ... × 99 × 102.

• A: 4
• B: 6
• C: 7
• D: 10

Correct Answer: (c)

Solution: Count factors of 5: 15, 30, 45, 60, 75, 90 → 6 zeros. The correct answer is (c).

Question 183: The unit's digit of 13²⁰⁰³ is

• A: 1
• B: 3
• C: 7
• D: 9

Correct Answer: (c)

Solution: The unit's digit of powers of 13 cycles as 3, 9, 7, 1. 2003 ÷ 4 = 500 remainder 3 → unit's digit is 7. The correct answer is (c).

Question 184: The digit in the unit's place of the number 123⁴⁵⁶ is

• A: 1
• B: 3
• C: 6
• D: 9

Correct Answer: (a)

Solution: The unit's digit of powers of 3 cycles as 3, 9, 7, 1. 456 ÷ 4 = 114 remainder 0 → unit's digit is 1. The correct answer is (a).

Question 185: Match List I with List II and select the correct answer:

• A: 1 4 3
• B: 4 2 3
• C: 4 2 5
• D: None of these

Correct Answer: (b)

Solution: Match the unit's digits based on the given options. The correct answer is (b).

Question 186: The digit in the unit's place of the number (61)²⁵ - 1 is

• A: 0
• B: 2
• C: 6
• D: None of these

Correct Answer: (a)

Solution: The unit's digit of powers of 61 is always 1. Subtracting 1 gives 0. The correct answer is (a).

Question 187: The unit's digit in the product 274 × 318 × 577 × 313 is

• A: 2
• B: 4
• C: 6
• D: 8

Correct Answer: (a)

Solution: Multiply unit's digits: 4 × 8 × 7 × 3 = 672 → unit's digit is 2. The correct answer is (a).

Question 188: In the product 459 × 46 × 28* × 484, the digit in the unit place is 8. The digit to come in place of * is

• A: 3
• B: 5
• C: 7
• D: None of these

Correct Answer: (b)

Solution: Multiply unit's digits: 9 × 6 × * × 4 = 8 → * = 5. The correct answer is (b).

Question 189: The digit in the unit place of the number represented by (7ⁿ - 3ˢ⁸) is

• A: 0
• B: 6
• C: 8
• D: None of these

Correct Answer: (a)

Solution: Unit's digit of 7ⁿ cycles as 7, 9, 3, 1. Unit's digit of 3ˢ⁸ is 9. Difference: 1 - 9 = 0. The correct answer is (a).

Question 190: The unit's digit in (784)¹²⁶ + (784)¹²⁷ is

• A: 0
• B: 2
• C: 6
• D: 8

Correct Answer: (a)

Solution: Unit's digit of 4¹²⁶ is 6, and 4¹²⁷ is 4. Sum: 6 + 4 = 10 → unit's digit is 0. The correct answer is (a).

Question 191: The digit in the unit's place of [(251)⁹⁸ + (21)²⁹ - (106)¹⁰⁰ + (705)³⁵ - 164 + 259] is

• A: 1
• B: 4
• C: 5
• D: 6

Correct Answer: (c)

Solution: Compute unit's digits: 1 + 1 - 6 + 5 - 4 + 9 = 5. The correct answer is (c).

Question 192: The digit in the unit's place of the product (2464)¹⁷⁹³ × (615)³¹⁷ × (131)⁴⁹¹ is

• A: 0
• B: 2
• C: 3
• D: 5

Correct Answer: (a)

Solution: Multiply unit's digits: 4 × 5 × 1 = 20 → unit's digit is 0. The correct answer is (a).

Question 193: If x is an even number, then xⁿ, where n is a positive integer, will always have

• A: Zero in the unit's place
• B: 6 in the unit's place
• C: Either 0 or 6 in the unit's place
• D: None of these

Correct Answer: (c)

Solution: For even x, xⁿ ends in 0 or 6 depending on x. The correct answer is (c).

Question 194: In the product 459 × 46 × 28* × 484, the digit in the unit place is 8. The digit to come in place of * is

• A: 3
• B: 5
• C: 7
• D: None of these

Correct Answer: (b)

Solution: Multiply unit digits: 9 × 6 × * × 4 = 8 → * = 5. The correct answer is (b).

Question 195: The digit in the unit place of the number represented by (7ⁿ - 3ˢ⁸) is

• A: 0
• B: 4
• C: 6
• D: 8

Correct Answer: (a)

Solution: Unit's digit of powers of 7 cycles as 7, 9, 3, 1. Unit's digit of 3ˢ⁸ is 9. Difference: 1 - 9 = 0. The correct answer is (a).

Question 196: Unit's digit in (784)¹²⁶ + (784)¹²⁷ is

• A: 0
• B: 2
• C: 6
• D: 8

Correct Answer: (a)

Solution: Unit's digit of 4¹²⁶ is 6, and 4¹²⁷ is 4. Sum: 6 + 4 = 10 → unit's digit is 0. The correct answer is (a).

Question 197: The digit in the unit's place of [(251)⁹⁸ + (21)²⁹ - (106)¹⁰⁰ + (705)³⁵ - 164 + 259] is

• A: 1
• B: 4
• C: 5
• D: 6

Correct Answer: (c)

Solution: Compute unit's digits: 1 + 1 - 6 + 5 - 4 + 9 = 5. The correct answer is (c).

Question 198: The digit in the unit's place of the product (2464)¹⁷⁹³ × (615)³¹⁷ × (131)⁴⁹¹ is

• A: 0
• B: 2
• C: 3
• D: 5

Correct Answer: (a)

Solution: Multiply unit's digits: 4 × 5 × 1 = 20 → unit's digit is 0. The correct answer is (a).

Question 199: If x is an even number, then xⁿ, where n is a positive integer, will always have

• A: Zero in the unit's place
• B: 6 in the unit's place
• C: Either 0 or 6 in the unit's place
• D: None of these

Correct Answer: (c)

Solution: For even x, xⁿ ends in 0 or 6 depending on x. The correct answer is (c).

Question 200: If m and n are positive integers, then the digit in the unit's place of 5ᵐ + 6ⁿ is always

• A: 1
• B: 5
• C: 6
• D: m + n

Correct Answer: (a)

Solution: Unit's digit of 5ᵐ is always 5, and 6ⁿ is always 6. Sum: 5 + 6 = 11 → unit's digit is 1. The correct answer is (a).

Question 201: The number formed from the last two digits (ones and tens) of the expression 212ⁿ - 64ⁿ, where n is any positive integer is

• A: 10
• B: 0
• C: 30
• D: 2

Correct Answer: (b)

Solution: Last two digits of 212ⁿ - 64ⁿ are always 00. The correct answer is (b).

Question 202: The last digit in the decimal representation of √2 is

• A: 2
• B: 4
• C: 5
• D: 6

Correct Answer: (c)

Solution: √2 is irrational, so its decimal representation does not terminate or repeat. The correct answer is (c).

Question 203: Let x be the product of two numbers 3,659,893,456,789,325,678 and 342,973,489,379,256. The number of digits in x is

• A: 32
• B: 34
• C: 35
• D: 36

Correct Answer: (d)

Solution: Number of digits in product = sum of digits in factors = 18 + 18 = 36. The correct answer is (d).

Question 204: Let a number of three digits have for its middle digit the sum of the other two digits. Then it is a multiple of

• A: 10
• B: 11
• C: 18
• D: 50

Correct Answer: (b)

Solution: Such numbers are divisible by 11. The correct answer is (b).

Question 205: What least value must be given to n so that the number 6135H2 becomes divisible by 9?

• A: 1
• B: 2
• C: 3
• D: 4

Correct Answer: (a)

Solution: Sum of digits = 6 + 1 + 3 + 5 + H + 2 = 17 + H. For divisibility by 9, H = 1. The correct answer is (a).

Question 206: Find the multiple of 11 in the following numbers.

• A: 112144
• B: 447355
• C: 869756
• D: 978626

Correct Answer: (c)

Solution: Check divisibility by 11: 869756 → (8 + 9 + 5) - (6 + 7 + 6) = 0. The correct answer is (c).

Question 207: 111,111,111,111 is divisible by

• A: 3 and 37 only
• B: 3, 11 and 37 only
• C: 3, 11, 37 and 111 only
• D: 3, 11, 37, 111 and 1001

Correct Answer: (d)

Solution: 111,111,111,111 is divisible by all listed numbers. The correct answer is (d).

Question 208: Which of the following numbers is not divisible by 18?

• A: 34056
• B: 50436
• C: 54036
• D: 65043

Correct Answer: (d)

Solution: Divisibility by 18 requires divisibility by both 2 and 9. 65043 is odd and not divisible by 2. The correct answer is (d).

Question 209: The number 89715938* is divisible by 4. The unknown non-zero digit marked as * will be

• A: 2
• B: 3
• C: 4
• D: 6

Correct Answer: (a)

Solution: Last two digits must be divisible by 4. 8* → * = 2. The correct answer is (a).

Question 210: Which one of the following numbers is divisible by 3?

• A: 4006020
• B: 2345678
• C: 2876423
• D: 9566003

Correct Answer: (a)

Solution: Sum of digits: 4006020 → 12 (divisible by 3). The correct answer is (a).

Question 211: A number is divisible by 11 if the difference between the sums of the digits in odd and even places respectively is

• A: A multiple of 3
• B: A multiple of 5
• C: Zero or a multiple of 7
• D: Zero or a multiple of 11

Correct Answer: (d)

Solution: Divisibility rule for 11. The correct answer is (d).

Question 212: Which one of the following numbers is divisible by 11?

• A: 4823718
• B: 4832718
• C: 8423718
• D: 8432718

Correct Answer: (d)

Solution: Check divisibility by 11: 8432718 → (8 + 3 + 7 + 8) - (4 + 2 + 1) = 0. The correct answer is (d).

Question 213: Which one of the following numbers is divisible by 15?

• A: 17325
• B: 23755
• C: 29515
• D: 30560

Correct Answer: (a)

Solution: Divisibility by 15 requires divisibility by both 3 and 5. 17325 is divisible by both. The correct answer is (a).

Question 214: 7386038 is divisible by

• A: 3
• B: 4
• C: 9
• D: 11

Correct Answer: (b)

Solution: Divisibility by 4: Last two digits 38 are divisible by 4. The correct answer is (b).

Question 215: The numbers 24984, 26784 and 28584 are

• A: Divisible by 3 and 4
• B: Divisible by 4 and 9
• C: Divisible by 3 and 9
• D: Divisible by 3, 4 and 9

Correct Answer: (a)

Solution: All numbers are divisible by 3 and 4 but not 9. The correct answer is (a).

Question 216: Which of the following numbers is a multiple of 8?

• A: 923872
• B: 923972
• C: 923862
• D: 923962

Correct Answer: (a)

Solution: Check divisibility by 8: Last three digits 872 are divisible by 8. The correct answer is (a).

Question 217: If 78*3945 is divisible by 11, where * is a digit, then * is equal to

• A: 0
• B: 1
• C: 3
• D: 5

Correct Answer: (c)

Solution: Check divisibility by 11: (7 + * + 9 + 5) - (8 + 3 + 4) = 0 → * = 3. The correct answer is (c).

Question 218: If m and n are integers divisible by 5, which of the following is not necessarily true?

• A: m + n is divisible by 10
• B: m - n is divisible by 5
• C: m² - n² is divisible by 25
• D: None of these

Correct Answer: (a)

Solution: m + n is not necessarily divisible by 10. The correct answer is (a).

Question 219: An integer is divisible by 16 if and only if its last X digits are divisible by 16. The value of X would be

• A: 3
• B: 4
• C: 5
• D: 6

Correct Answer: (b)

Solution: Last 4 digits determine divisibility by 16. The correct answer is (b).

Question 220: Which of the following numbers is divisible by 3, 7, 9 and 11?

• A: 639
• B: 2079
• C: 3791
• D: 37911

Correct Answer: (b)

Solution: Check divisibility: 2079 is divisible by all. The correct answer is (b).

Question 221: A number 476**0 is divisible by both 3 and 11. The non-zero digits in the hundred's and ten's place respectively are

• A: 7, 4
• B: 5, 3
• C: 5, 2
• D: None of these

Correct Answer: (c)

Solution: Solve using divisibility rules: 476520 satisfies both conditions. The correct answer is (c).

Question 222: How many of the following numbers are divisible by 3 but not by 9?

• A: 5
• B: 6
• C: 7
• D: None of these

Correct Answer: (a)

Solution: Numbers divisible by 3 but not 9: 5. The correct answer is (a).

Question 223: If the number 357*25* is divisible by both 3 and 5, then the missing digits in the unit's place and the thousandth's place respectively are

• A: 0, 6
• B: 5, 1
• C: 5, 4
• D: None of these

Correct Answer: (b)

Solution: Solve using divisibility rules: 357525 satisfies both conditions. The correct answer is (b).

Question 224: 6897 is divisible by

• A: 11 only
• B: 19 only
• C: Both 11 and 19
• D: Neither 11 nor 19

Correct Answer: (c)

Solution: Check divisibility: 6897 is divisible by both 11 and 19. The correct answer is (c).

Question 225: Which of the following numbers is exactly divisible by 24?

• A: 35718
• B: 63810
• C: 537804
• D: 3125736

Correct Answer: (d)

Solution: Divisibility by 24 requires divisibility by both 3 and 8. 3125736 satisfies both. The correct answer is (d).

Question 226: The number is

• A: Neither divisible by 3 nor by 11
• B: Divisible by 11 but not by 3
• C: Divisible by 3 but not by 11
• D: Divisible by both 3 and 11

Correct Answer: (d)

Solution: Check divisibility rules for 3 and 11. The correct answer is (d).

Question 227: 325325 is a six-digit number. It is divisible by

• A: 7 only
• B: 11 only
• C: 13 only
• D: All 7, 11, and 13

Correct Answer: (d)

Solution: 325325 is divisible by 7, 11, and 13. The correct answer is (d).

Question 228: If the seven-figure number 30X0103 is a multiple of 13, then X is

• A: 1
• B: 6
• C: 7
• D: 8

Correct Answer: (c)

Solution: Solve using divisibility rule for 13. X = 7 satisfies the condition. The correct answer is (c).

Question 229: If a number is divisible by both 11 and 13, then it must be necessarily

• A: 429
• B: Divisible by (11 × 13)
• C: Divisible by (13 - 11)
• D: None of these

Correct Answer: (b)

Solution: A number divisible by both 11 and 13 is divisible by their product. The correct answer is (b).

Question 230: Which of the following numbers are completely divisible by 7?

• A: Only I and II
• B: Only II and III
• C: Only II and IV
• D: All are divisible

Correct Answer: (d)

Solution: Check divisibility by 7 for all numbers. All are divisible. The correct answer is (d).

Question 231: If x and y are two digits of the number 653xy such that the number is divisible by 80, then x + y is equal to

• A: 3
• B: 4
• C: 5
• D: 6

Correct Answer: (b)

Solution: For divisibility by 80, last three digits must be divisible by 80. x + y = 4. The correct answer is (b).

Question 232: The six-digit number 5ABB7A is a multiple of 33 for non-zero digits A and B. Which of the following could be possible value of A + B?

• A: 8
• B: 9
• C: 10
• D: 14

Correct Answer: (a)

Solution: Solve using divisibility rules for 3 and 11. A + B = 8 satisfies the condition. The correct answer is (a).

Question 233: Which of the following numbers is divisible by 99?

• A: 114345
• B: 913464
• C: 135792
• D: 3572404

Correct Answer: (a)

Solution: Check divisibility by 99 (both 9 and 11). 114345 is divisible by 99. The correct answer is (a).

Question 234: The digits indicated by * in 3422213** so that this number is divisible by 99 are

• A: 1, 9
• B: 3, 7
• C: 4, 6
• D: 5, 5

Correct Answer: (c)

Solution: Solve using divisibility rules for 9 and 11. * = 4, 6 satisfies the condition. The correct answer is (c).

Question 235: If 37X3 is a four-digit natural number divisible by 7, then the place marked as X must have the value

• A: 0
• B: 3
• C: 5
• D: 9

Correct Answer: (b)

Solution: Check divisibility by 7. X = 3 satisfies the condition. The correct answer is (b).

Question 236: If the seven-digit number 876p37q is divisible by 225, then the values of p and q respectively are

• A: 0 and 0
• B: 9 and 0
• C: 0 and 5
• D: 9 and 5

Correct Answer: (c)

Solution: For divisibility by 225, the number must be divisible by both 9 and 25. p = 0, q = 5 satisfies the condition. The correct answer is (c).

Question 237: If a number 774958A96B is divisible by 8 and 9, the respective values of A and B will be

• A: 5 and 8
• B: 7 and 8
• C: 8 and 0
• D: None of these

Correct Answer: (c)

Solution: Check divisibility by 8 and 9. A = 8, B = 0 satisfies the condition. The correct answer is (c).

Question 238: How many of the following numbers are divisible by 132?

• A: 4
• B: 5
• C: 6
• D: 7

Correct Answer: (a)

Solution: Check divisibility by 132 (both 12 and 11). 4 numbers satisfy the condition. The correct answer is (a).

Question 239: If x and y are positive integers such that (3x + 7y) is a multiple of 11, then which of the following is also a multiple of 11?

• A: 5x - 3y
• B: 9x + 4y
• C: 4x + 6y
• D: x + y + 6

Correct Answer: (a)

Solution: Solve using modular arithmetic. 5x - 3y is a multiple of 11. The correct answer is (a).

Question 240: If n be any natural number then by which largest number (n³ - n) is always divisible?

• A: 3
• B: 6
• C: 12
• D: 18

Correct Answer: (b)

Solution: Factorize n³ - n = n(n² - 1) = n(n - 1)(n + 1). Product of three consecutive integers is always divisible by 6. The correct answer is (b).

Question 241: If a and b are two odd positive integers, by which of the following integers is (a⁴ - b⁴) always divisible?

• A: 3
• B: 6
• C: 8
• D: 12

Correct Answer: (c)

Solution: Simplify a⁴ - b⁴ = (a² + b²)(a² - b²). Both terms are divisible by 8 for odd integers. The correct answer is (c).

Question 242: The difference between the squares of any two consecutive integers is equal to

• A: An even number
• B: Difference of given numbers
• C: Sum of given numbers
• D: Product of given numbers

Correct Answer: (c)

Solution: Let integers be n and n+1. Difference = (n+1)² - n² = 2n + 1 = sum of integers. The correct answer is (c).

Question 243: The number 6n² + 6n for natural number n is always divisible by

• A: 6 only
• B: 6 and 12
• C: 12 only
• D: 18 only

Correct Answer: (b)

Solution: Factorize 6n² + 6n = 6n(n + 1). Always divisible by 6 and 12. The correct answer is (b).

Question 244: The difference of a number consisting of two digits and the number formed by interchanging the digits is always divisible by

• A: 5
• B: 9
• C: 11
• D: None of these

Correct Answer: (b)

Solution: Difference = 9 × (difference of digits). Always divisible by 9. The correct answer is (b).

Question 245: The sum of a number consisting of two digits and the number formed by interchanging the digits is always divisible by

• A: 7
• B: 9
• C: 10
• D: 11

Correct Answer: (a)

Solution: Sum = 11 × (sum of digits). Always divisible by 11. The correct answer is (a).

Question 246: The largest natural number, which exactly divides the product of any four consecutive natural numbers, is

• A: 6
• B: 12
• C: 24
• D: 120

Correct Answer: (c)

Solution: Product of four consecutive numbers is divisible by 24. The correct answer is (c).

Question 247: If n is a whole number greater than 1, then n²(n² - 1) is always divisible by

• A: 8
• B: 10
• C: 12
• D: 16

Correct Answer: (c)

Solution: Factorize n²(n² - 1) = n²(n - 1)(n + 1). Always divisible by 12. The correct answer is (c).

Question 248: If n is any odd number greater than 1, then n(n² - 1) is always divisible by

• A: 3
• B: 6
• C: 12
• D: 18

Correct Answer: (b)

Solution: Factorize n(n² - 1) = n(n - 1)(n + 1). Always divisible by 6. The correct answer is (b).

Question 249: The difference between the squares of two consecutive odd integers is always divisible by

• A: 3
• B: 6
• C: 7
• D: 8

Correct Answer: (d)

Solution: Difference = 8 × (sum of integers). Always divisible by 8. The correct answer is (d).

Question 250: The smallest 4-digit number exactly divisible by 7 is

• A: 1001
• B: 1007
• C: 1101
• D: 1108

Correct Answer: (a)

Solution: Divide 1000 by 7, find the next multiple. Smallest is 1001. The correct answer is (a).

Question 251: What least number must be added to 1056 to get a number exactly divisible by 23?

• A: 2
• B: 3
• C: 21
• D: 25

Correct Answer: (a)

Solution: Divide 1056 by 23, find remainder. Add (23 - remainder). Add 2. The correct answer is (a).

Question 252: Which of the following numbers should be added to 8567 to make it exactly divisible by 4?

• A: 3
• B: 4
• C: 5
• D: 6

Correct Answer: (a)

Solution: Check divisibility by 4. Add 3. The correct answer is (a).

Question 253: The difference between the squares of two consecutive even integers is divisible by

• A: 3
• B: 4
• C: 6
• D: 7

Correct Answer: (b)

Solution: Difference = 4 × (sum of integers). Always divisible by 4. The correct answer is (b).

Question 254: The difference between the squares of two consecutive odd integers is divisible by

• A: 3
• B: 6
• C: 7
• D: 8

Correct Answer: (d)

Solution: Difference = 8 × (sum of integers). Always divisible by 8. The correct answer is (d).

Question 255: The difference between the squares of two consecutive odd integers is always divisible by

• A: 3
• B: 6
• C: 7
• D: 8

Correct Answer: (d)

Solution: Difference = 8 × (sum of integers). Always divisible by 8. The correct answer is (d).

Question 256: A 4-digit number is formed by repeating a 2-digit number such as 2525, 3232 etc. Any number of this form is exactly divisible by

• A: 7
• B: 11
• C: 13
• D: Smallest 3-digit prime number

Correct Answer: (d)

Solution: Numbers like 2525 are divisible by 101 (smallest 3-digit prime). The correct answer is (d).

Question 257: A 6-digit number is formed by repeating a 3-digit number; for example, 256256 or 678678 etc. Any number of this form is always exactly divisible by

• A: 7 only
• B: 11 only
• C: 13 only
• D: 1001

Correct Answer: (d)

Solution: Numbers like 256256 are divisible by 1001. The correct answer is (d).

Question 258: The sum of the digits of a natural number (10ⁿ - 1) is 4707, where n is a natural number. The value of n is

• A: 477
• B: 523
• C: 532
• D: 704

Correct Answer: (b)

Solution: Sum of digits = 9 × n. Solve for n. n = 523. The correct answer is (b).

Question 259: (xⁿ - aⁿ) is divisible by (x - a)

• A: For all values of n
• B: Only for even values of n
• C: Only for odd values of n
• D: Only for prime values of n

Correct Answer: (a)

Solution: (xⁿ - aⁿ) is divisible by (x - a) for all n. The correct answer is (a).

Question 260: Which one of the following is the number by which the product of 8 consecutive integers is divisible?

• A: 4!
• B: 6!
• C: 7!
• D: All of these

Correct Answer: (d)

Solution: Product of 8 consecutive integers is divisible by 8!, which includes 4!, 6!, and 7!. The correct answer is (d).

Question 261: Consider the following statements: For any positive integer n, the number 10ⁿ - 1 is divisible by

• A: 9 for n = odd only
• B: 9 for n = even only
• C: 11 for n = odd only
• D: 11 for n = even only

Correct Answer: (a)

Solution: 10ⁿ - 1 is divisible by 9 for all n, and by 11 for odd n. The correct answer is (a).

Question 262: If n is any positive integer, 3⁴ⁿ - 4³ⁿ is always divisible by

• A: 7
• B: 12
• C: 17
• D: 145

Correct Answer: (a)

Solution: Simplify 3⁴ⁿ - 4³ⁿ. Always divisible by 7. The correct answer is (a).

Question 263: If the square of an odd natural number is divided by 8, then the remainder will be

• A: 1
• B: 2
• C: 3
• D: 4

Correct Answer: (a)

Solution: Square of odd number = 8k + 1. Remainder is 1. The correct answer is (a).

Question 264: The largest number that exactly divides each number of the sequence l⁵ - 1, 2⁵ - 2, 3⁵ - 3, ..., n⁵ - n,.... is

• A: 1
• B: 15
• C: 30
• D: 120

Correct Answer: (c)

Solution: Factorize n⁵ - n = n(n⁴ - 1) = n(n² - 1)(n² + 1). For all n, this is divisible by 30. The correct answer is (c).

Question 265: The difference of the squares of two consecutive even integers is divisible by

• A: 3
• B: 4
• C: 6
• D: 7

Correct Answer: (b)

Solution: Difference = 4 × (sum of integers). Always divisible by 4. The correct answer is (b).

Question 266: The difference of the squares of two consecutive odd integers is divisible by

• A: 3
• B: 6
• C: 7
• D: 8

Correct Answer: (d)

Solution: Difference = 8 × (sum of integers). Always divisible by 8. The correct answer is (d).

Question 267: The smallest 4-digit number exactly divisible by 7 is

• A: 1001
• B: 1007
• C: 1101
• D: 1108

Correct Answer: (a)

Solution: Divide 1000 by 7, find the next multiple. Smallest is 1001. The correct answer is (a).

Question 268: What least number must be added to 1056 to get a number exactly divisible by 23?

• A: 2
• B: 3
• C: 21
• D: 25

Correct Answer: (a)

Solution: Divide 1056 by 23, find remainder. Add (23 - remainder). Add 2. The correct answer is (a).

Question 269: Which of the following numbers should be added to 8567 to make it exactly divisible by 4?

• A: 3
• B: 4
• C: 5
• D: 6

Correct Answer: (a)

Solution: Check divisibility by 4. Add 3. The correct answer is (a).

Question 270: If all the numbers from 501 to 700 are written, what is the total number of times the digit 6 appears?

• A: 138
• B: 139
• C: 140
• D: 141

Correct Answer: (c)

Solution: Count occurrences of 6 in hundreds, tens, and units places. Total = 140. The correct answer is (c).

Question 271: How many numbers less than 1000 are multiples of both 10 and 13?

• A: 6
• B: 7
• C: 8
• D: 9

Correct Answer: (b)

Solution: LCM(10, 13) = 130. Numbers = 130, 260, ..., 910. Total = 7. The correct answer is (b).

Question 272: How many integers between 100 and 150, both inclusive, can be evenly divided by neither 3 nor 5?

• A: 26
• B: 27
• C: 28
• D: 33

Correct Answer: (b)

Solution: Total numbers = 51. Exclude multiples of 3 and 5 using inclusion-exclusion. Remaining = 27. The correct answer is (b).

Question 273: A number when divided by the sum of 555 and 445 gives two times their difference as quotient and 30 as the remainder. The number is

• A: 1220
• B: 1250
• C: 22030
• D: 220030

Correct Answer: (d)

Solution: Solve using division formula. Number = 220030. The correct answer is (d).

Question 274: In doing a question of division with zero remainder, a candidate took 12 as divisor instead of 21. The quotient obtained by him was 35. The correct quotient is

• A: 0
• B: 12
• C: 13
• D: 20

Correct Answer: (d)

Solution: Solve using division formula. Correct quotient = 20. The correct answer is (d).

Question 275: A number when divided by 19 leaves a remainder 9. The remainder when the square of the number is divided by 19 is

• A: 1
• B: 3
• C: 5
• D: 9

Correct Answer: (a)

Solution: Square of remainder = 9² = 81. Remainder when 81 is divided by 19 = 1. The correct answer is (a).

Question 276: When n is divided by 4, the remainder is 3. What is the remainder when 2n is divided by 4?

• A: 1
• B: 2
• C: 3
• D: 6

Correct Answer: (b)

Solution: 2n = 2 × (4k + 3) = 8k + 6. Remainder = 2. The correct answer is (b).

Question 277: The number of times 99 is subtracted from 11001 so that the remainder is less than 99 is

• A: 110
• B: 111
• C: 112
• D: 113

Correct Answer: (b)

Solution: Divide 11001 by 99. Quotient = 111. The correct answer is (b).

Question 278: When a number is divided by 13, the remainder is 11. When the same number is divided by 17, the remainder is 9. What is the number?

• A: 339
• B: 349
• C: 369
• D: Data inadequate

Correct Answer: (b)

Solution: Solve using simultaneous congruences. Number = 349. The correct answer is (b).

Question 279: In a division sum, the remainder was 71. With the same divisor but twice the dividend, the remainder is 43. Which one of the following is the divisor?

• A: 86
• B: 93
• C: 99
• D: 104

Correct Answer: (c)

Solution: Solve using division formula. Divisor = 99. The correct answer is (c).

Question 280: A number when divided by 3 leaves a remainder 1. When the quotient is divided by 2, it leaves a remainder 1. What will be the remainder when the number is divided by 6?

• A: 2
• B: 3
• C: 4
• D: 5

Correct Answer: (d)

Solution: Solve step-by-step. Number = 6k + 5. Remainder = 5. The correct answer is (d).

Question 281: When the square of any odd number, greater than 1, is divided by 8, it always leaves remainder

• A: 1
• B: 6
• C: 8
• D: Cannot be determined

Correct Answer: (a)

Solution: Square of odd number = 8k + 1. Remainder = 1. The correct answer is (a).

Question 282: The numbers from 1 to 29 are written side by side. If this number is divided by 9, then what is the remainder?

• A: 0
• B: 1
• C: 3
• D: None of these

Correct Answer: (c)

Solution: Sum of digits = 435. Remainder when 435 is divided by 9 = 3. The correct answer is (c).

Question 283: If 17²⁰⁰ is divided by 18, the remainder is

• A: 1
• B: 2
• C: 16
• D: 17

Correct Answer: (a)

Solution: Use modular arithmetic. Remainder = 1. The correct answer is (a).

Question 284: What is the remainder when 2³¹ is divided by 5?

• A: 0
• B: 1
• C: 2
• D: 3

Correct Answer: (d)

Solution: Use modular arithmetic. Remainder = 3. The correct answer is (d).

Question 285: Consider the following statements: (1) aⁿ + bⁿ is divisible by a + b if n = 2k + 1. (2) aⁿ - bⁿ is divisible by a - b if n = 2k. Which of the statements given above is/are correct?

• A: 1 only
• B: 2 only
• C: Both 1 and 2
• D: Neither 1 nor 2

Correct Answer: (c)

Solution: Both statements are true based on divisibility rules. The correct answer is (c).

Question 286: (7¹⁹ + 2) is divided by 6. The remainder is

• A: 0
• B: 1
• C: 2
• D: 3

Correct Answer: (b)

Solution: Simplify using modular arithmetic. Remainder = 1. The correct answer is (b).

Question 287: If (10¹² + 25)² - (10¹² - 25)² = 10ⁿ, then the value of n is

• A: 10
• B: 12
• C: 15
• D: 20

Correct Answer: (b)

Solution: Simplify using algebraic identity. n = 12. The correct answer is (b).

Question 288: (3²⁵ + 3²⁶ + 3²⁷ + 3²⁸) is divisible by

• A: 11
• B: 16
• C: 25
• D: 30

Correct Answer: (d)

Solution: Factorize and simplify. Divisible by 30. The correct answer is (d).

Question 289: (4⁶¹ + 4⁶² + 4⁶³ + 4⁶⁴) is divisible by

• A: 11
• B: 13
• C: 17
• D: 19

Correct Answer: (a)

Solution: Factorize and simplify. Divisible by 11. The correct answer is (a).

Question 290: (9⁶ + 1) when divided by 8, would leave a remainder of

• A: 0
• B: 1
• C: 2
• D: 3

Correct Answer: (b)

Solution: Simplify using modular arithmetic. Remainder = 1. The correct answer is (b).

Question 291: If n is even, (6ⁿ - 1) is divisible by

• A: 5
• B: 7
• C: 35
• D: 49

Correct Answer: (c)

Solution: Simplify using modular arithmetic. Divisible by 35. The correct answer is (c).

Question 292: 2⁵¹⁵ is divided by 26, the remainder is

• A: 1
• B: 2
• C: 24
• D: 25

Correct Answer: (d)

Solution: Simplify using modular arithmetic. Remainder = 25. The correct answer is (d).

Question 293: If (67⁶⁷ + 67) is divided by 68, the remainder is

• A: 0
• B: 1
• C: 66
• D: 67

Correct Answer: (d)

Solution: Simplify using modular arithmetic. Remainder = 67. The correct answer is (d).

Question 294: One less than (49)¹⁵ is exactly divisible by

• A: 2
• B: 3
• C: 5
• D: 7

Correct Answer: (b)

Solution: Simplify using modular arithmetic. Divisible by 3. The correct answer is (b).

Question 295: The remainder when 7⁸⁴ is divided by 342 is

• A: 0
• B: 1
• C: 7
• D: 49

Correct Answer: (b)

Solution: Simplify using modular arithmetic. Remainder = 1. The correct answer is (b).

Question 296: The remainder when 2⁶⁰ is divided by 5 equals

• A: 1
• B: 2
• C: 3
• D: 4

Correct Answer: (a)

Solution: Simplify using modular arithmetic. Remainder = 1. The correct answer is (a).

Question 297: By how many of the following numbers is 2¹² - 1 divisible? 2, 3, 5, 7, 10, 11, 13, 14

• A: 4
• B: 5
• C: 6
• D: 7

Correct Answer: (c)

Solution: Factorize 2¹² - 1. Divisible by 6 numbers. The correct answer is (c).

Question 298: The remainder when (15²³ + 23³³) is divided by 19 is

• A: 0
• B: 1
• C: 13
• D: 17

Correct Answer: (a)

Solution: Simplify using modular arithmetic. Remainder = 0. The correct answer is (a).

Question 299: When 2²⁵⁶ is divided by 17, the remainder would be

• A: 1
• B: 13
• C: 15
• D: 17

Correct Answer: (a)

Solution: Simplify using modular arithmetic. Remainder = 1. The correct answer is (a).

Question 300: 7⁶ⁿ - 6⁶ⁿ, where n is an integer > 0, is divisible by

• A: 13
• B: 127
• C: 559
• D: All of these

Correct Answer: (d)

Solution: Simplify using modular arithmetic. Divisible by all. The correct answer is (d).

Question 301: In a division problem, the divisor is 7 times of quotient and 5 times of remainder. If the dividend is 6 times of remainder, then the quotient is equal to

• A: 0
• B: 1
• C: 7
• D: None of these

Correct Answer: (b)

Solution: Solve using relationships between divisor, quotient, and remainder. Quotient = 1. The correct answer is (b).

Question 302: On dividing a number by 19, the difference between quotient and remainder is 9. The number is

• A: 352
• B: 361
• C: 370
• D: 371

Correct Answer: (c)

Solution: Solve using division formula. Number = 370. The correct answer is (c).

Question 303: A number when divided by 136 leaves remainder 36. If the same number is divided by 17, the remainder will be

• A: 2
• B: 3
• C: 7
• D: 9

Correct Answer: (a)

Solution: Simplify using modular arithmetic. Remainder = 2. The correct answer is (a).

Question 304: A number when divided by 195 leaves a remainder 47. If the same number is divided by 15, the remainder will be

• A: 1
• B: 2
• C: 3
• D: 4

Correct Answer: (c)

Solution: Simplify using modular arithmetic. Remainder = 3. The correct answer is (c).

Question 305: A certain number when divided by 899 gives a remainder 63. What is the remainder when the same number is divided by 29?

• A: 5
• B: 25
• C: 27
• D: None of these

Correct Answer: (a)

Solution: Simplify using modular arithmetic. Remainder = 5. The correct answer is (a).

Question 306: A number when divided by 5 leaves the remainder 3. What is the remainder when the square of the same number is divided by 5?

• A: 0
• B: 3
• C: 4
• D: 9

Correct Answer: (c)

Solution: Square of remainder = 9. Remainder when divided by 5 = 4. The correct answer is (c).

Question 307: The difference between two numbers is 1365. When the larger number is divided by the smaller one, the quotient is 6 and the remainder is 15. What is the smaller number?

• A: 240
• B: 270
• C: 295
• D: 360

Correct Answer: (d)

Solution: Solve using division formula. Smaller number = 360. The correct answer is (d).

Question 308: When n is divided by 4, the remainder is 3. What is the remainder when 2n is divided by 4?

• A: 1
• B: 2
• C: 3
• D: 6

Correct Answer: (b)

Solution: 2n = 2 × (4k + 3) = 8k + 6. Remainder = 2. The correct answer is (b).

Question 309: When a number is divided by 13, the remainder is 11. When the same number is divided by 17, the remainder is 9. What is the number?

• A: 339
• B: 349
• C: 369
• D: Data inadequate

Correct Answer: (b)

Solution: Solve using simultaneous congruences. Number = 349. The correct answer is (b).

Question 310: In a division sum, the remainder was 71. With the same divisor but twice the dividend, the remainder is 43. Which one of the following is the divisor?

• A: 86
• B: 93
• C: 99
• D: 104

Correct Answer: (c)

Solution: Solve using division formula. Divisor = 99. The correct answer is (c).

Question 311: A positive integer P is divided by another positive integer, leaving remainder r1. Another positive integer Q is divided by the same integer, leaving remainder r2. When (P + Q) is divided by the same divisor, the remainder is r3. Which of the following could be the divisor?

• A: r1r2r3
• B: r1 + r2 + r3
• C: r1 - r2 + r3
• D: r1 + r2 - r3
• E: Cannot be determined

Correct Answer: (d)

Solution: Divisor could be r1 + r2 - r3. The correct answer is (d).

Question 312: Two numbers when divided by a certain divisor leave remainders 4375 and 2986 respectively. When the sum of the two numbers is divided by the same divisor, the remainder is 2361. The divisor in question is

• A: 4675
• B: 4900
• C: 5000
• D: None of these

Correct Answer: (b)

Solution: Solve using modular arithmetic. Divisor = 4900. The correct answer is (b).

Question 313: A number divided by 13 leaves a remainder 1 and if the quotient, thus obtained, is divided by 5, we get a remainder of 3. What will be the remainder if the number is divided by 65?

• A: 16
• B: 18
• C: 28
• D: 40

Correct Answer: (c)

Solution: Solve step-by-step. Remainder = 28. The correct answer is (c).

Question 314: The numbers 2272 and 875 are divided by a three-digit number N, giving the same remainder. The sum of the digits of N is

• A: 10
• B: 11
• C: 12
• D: 13

Correct Answer: (a)

Solution: Solve using modular arithmetic. Sum of digits = 10. The correct answer is (a).

Question 315: A number when divided by three consecutive numbers 9, 11, 13 leaves the remainders 8, 9, and 8 respectively. If the order of divisors is reversed, the remainders will be

• A: 10, 8, 9
• B: 10, 1, 6
• C: 8, 9, 8
• D: 9, 8, 8

Correct Answer: (b)

Solution: Solve using modular arithmetic. Remainders = 10, 1, 6. The correct answer is (b).

Question 316: After the division of a number successively by 3, 4, and 7, the remainders obtained are 2, 1, and 4 respectively. What will be the remainder if 84 divides the same number?

• A: 41
• B: 53
• C: 75
• D: 80

Correct Answer: (a)

Solution: Solve using modular arithmetic. Remainder = 41. The correct answer is (a).

Question 317: A number is successively divided by 8, 7, and 3 giving residues 3, 4, and 2 respectively and quotient 31. The number is

• A: 3555
• B: 5355
• C: 5535
• D: 5553

Correct Answer: (b)

Solution: Solve step-by-step. Number = 5355. The correct answer is (b).

Question 318: A number when divided by 3 leaves a remainder 1. When the quotient is divided by 2, it leaves a remainder 1. What will be the remainder when the number is divided by 6?

• A: 2
• B: 3
• C: 4
• D: 5

Correct Answer: (d)

Solution: Solve step-by-step. Remainder = 5. The correct answer is (d).

Question 319: When the square of any odd number, greater than 1, is divided by 8, it always leaves remainder

• A: 1
• B: 6
• C: 8
• D: Cannot be determined

Correct Answer: (a)

Solution: Square of odd number = 8k + 1. Remainder = 1. The correct answer is (a).

Question 320: The numbers from 1 to 29 are written side by side. If this number is divided by 9, then what is the remainder?

• A: 0
• B: 1
• C: 3
• D: None of these

Correct Answer: (c)

Solution: Sum of digits = 435. Remainder when 435 is divided by 9 = 3. The correct answer is (c).

Question 321: If 17²⁰⁰ is divided by 18, the remainder is

• A: 1
• B: 2
• C: 16
• D: 17

Correct Answer: (a)

Solution: Use modular arithmetic. Remainder = 1. The correct answer is (a).

Question 322: What is the remainder when 2³¹ is divided by 5?

• A: 0
• B: 1
• C: 2
• D: 3

Correct Answer: (d)

Solution: Use modular arithmetic. Remainder = 3. The correct answer is (d).

Question 323: Consider the following statements: (1) aⁿ + bⁿ is divisible by a + b if n = 2k + 1. (2) aⁿ - bⁿ is divisible by a - b if n = 2k. Which of the statements given above is/are correct?

• A: 1 only
• B: 2 only
• C: Both 1 and 2
• D: Neither 1 nor 2

Correct Answer: (c)

Solution: Both statements are true based on divisibility rules. The correct answer is (c).

Question 324: (7¹⁹ + 2) is divided by 6. The remainder is

• A: 0
• B: 1
• C: 2
• D: 3

Correct Answer: (b)

Solution: Simplify using modular arithmetic. Remainder = 1. The correct answer is (b).

Question 325: If (10¹² + 25)² - (10¹² - 25)² = 10ⁿ, then the value of n is

• A: 10
• B: 12
• C: 15
• D: 20

Correct Answer: (b)

Solution: Simplify using algebraic identity. n = 12. The correct answer is (b).

Question 326: (3²⁵ + 3²⁶ + 3²⁷ + 3²⁸) is divisible by

• A: 11
• B: 16
• C: 25
• D: 30

Correct Answer: (d)

Solution: Factorize and simplify. Divisible by 30. The correct answer is (d).

Question 327: (4⁶¹ + 4⁶² + 4⁶³ + 4⁶⁴) is divisible by

• A: 11
• B: 13
• C: 17
• D: 19

Correct Answer: (a)

Solution: Factorize and simplify. Divisible by 11. The correct answer is (a).

Question 328: (9⁶ + 1) when divided by 8, would leave a remainder of

• A: 0
• B: 1
• C: 2
• D: 3

Correct Answer: (b)

Solution: Simplify using modular arithmetic. Remainder = 1. The correct answer is (b).

Question 329: If n is even, (6ⁿ - 1) is divisible by

• A: 5
• B: 7
• C: 35
• D: 49

Correct Answer: (c)

Solution: Simplify using modular arithmetic. Divisible by 35. The correct answer is (c).

Question 330: 2⁵¹⁵ is divided by 26, the remainder is

• A: 1
• B: 2
• C: 24
• D: 25

Correct Answer: (d)

Solution: Simplify using modular arithmetic. Remainder = 25. The correct answer is (d).

Question 331: If (67⁶⁷ + 67) is divided by 68, the remainder is

• A: 0
• B: 1
• C: 66
• D: 67

Correct Answer: (d)

Solution: Simplify using modular arithmetic. Remainder = 67. The correct answer is (d).

Question 332: One less than (49)¹⁵ is exactly divisible by

• A: 2
• B: 3
• C: 5
• D: 7

Correct Answer: (b)

Solution: Simplify using modular arithmetic. Divisible by 3. The correct answer is (b).

Question 333: The remainder when 7⁸⁴ is divided by 342 is

• A: 0
• B: 1
• C: 7
• D: 49

Correct Answer: (b)

Solution: Simplify using modular arithmetic. Remainder = 1. The correct answer is (b).

Question 334: The remainder when 2⁶⁰ is divided by 5 equals

• A: 1
• B: 2
• C: 3
• D: 4

Correct Answer: (a)

Solution: Simplify using modular arithmetic. Remainder = 1. The correct answer is (a).

Question 335: By how many of the following numbers is 2¹² - 1 divisible? 2, 3, 5, 7, 10, 11, 13, 14

• A: 4
• B: 5
• C: 6
• D: 7

Correct Answer: (c)

Solution: Factorize 2¹² - 1. Divisible by 6 numbers. The correct answer is (c).

Question 336: The remainder when (15²³ + 23³³) is divided by 19 is

• A: 0
• B: 1
• C: 13
• D: 17

Correct Answer: (a)

Solution: Simplify using modular arithmetic. Remainder = 0. The correct answer is (a).

Question 337: When 2²⁵⁶ is divided by 17, the remainder would be

• A: 1
• B: 13
• C: 15
• D: 17

Correct Answer: (a)

Solution: Simplify using modular arithmetic. Remainder = 1. The correct answer is (a).

Question 338: 7⁶ⁿ - 6⁶ⁿ, where n is an integer > 0, is divisible by

• A: 13
• B: 127
• C: 559
• D: All of these

Correct Answer: (d)

Solution: Simplify using modular arithmetic. Divisible by all. The correct answer is (d).

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Correct Answer: nan

Solution: nan

Question 340: nan

Correct Answer: nan

Solution: nan

Question 341: nan

Correct Answer: nan

Solution: nan

Question 342: nan

Correct Answer: nan

Solution: nan

Question 343: nan

Correct Answer: nan

Solution: nan

Question 344: nan

Correct Answer: nan

Solution: nan

Question 345: Question

• A: Option A
• B: Option B
• C: Option C
• D: Option D
• E: Option E

Correct Answer: Correct Answer

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• A: -------------------
• B: -------------------
• C: -------------------
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Question 347: 7⁶ⁿ - 6⁶ⁿ, where n is an integer > 0, is divisible by

• A: 13
• B: 127
• C: 559
• D: All of these

Correct Answer: (d)

Solution: Simplify using modular arithmetic. Divisible by all. The correct answer is (d).

Question 348: It is given that (2³² + 1) is exactly divisible by a certain number. Which of the following is also definitely divisible by the same number?

• A: 2¹⁶ + 1
• B: 2¹⁶ - 1
• C: 7 × 2³³
• D: 2⁹⁶ + 1

Correct Answer: (d)

Solution: Simplify using modular arithmetic. Divisible by 2⁹⁶ + 1. The correct answer is (d).

Question 349: The number (2⁴⁸ - 1) is exactly divisible by two numbers between 60 and 70. The numbers are

• A: 63 and 65
• B: 63 and 67
• C: 61 and 65
• D: 65 and 67

Correct Answer: (a)

Solution: Solve using factorization. Numbers = 63 and 65. The correct answer is (a).

Question 350: n being any odd number greater than 1, n⁶⁵ - n is always divisible by

• A: 5
• B: 13
• C: 24
• D: None of these

Correct Answer: (c)

Solution: Use modular arithmetic. Divisible by 24. The correct answer is (c).

Question 351: Let N = 55³ + 17³ - 72³. Then, N is divisible by

• A: both 7 and 13
• B: both 3 and 13
• C: both 17 and 7
• D: both 3 and 17

Correct Answer: (b)

Solution: Factorize and simplify. Divisible by both 3 and 13. The correct answer is (b).

Question 352: Find the last two digits of N.

• A: 0
• B: 19
• C: 37
• D: 49

Correct Answer: (a)

Solution: Simplify using modular arithmetic. Last two digits = 00. The correct answer is (a).

Question 353: Find the remainder when N is divided by 168.

• A: 33
• B: 129
• C: 19
• D: 49

Correct Answer: (c)

Solution: Simplify using modular arithmetic. Remainder = 19. The correct answer is (c).

Question 354: What is the remainder when 4⁶¹ is divided by 51?

• A: 1
• B: 16
• C: 19
• D: 49

Correct Answer: (a)

Solution: Simplify using modular arithmetic. Remainder = 1. The correct answer is (a).

Question 355: What is the remainder when 17³⁶ is divided by 36?

• A: 1
• B: 16
• C: 19
• D: 49

Correct Answer: (a)

Solution: Simplify using modular arithmetic. Remainder = 1. The correct answer is (a).

Question 356: Which one of the following is the common factor of (47⁴³ + 43⁴³) and (47⁴⁷ + 43⁴⁷)?

• A: (47 - 43)
• B: (47 + 43)
• C: (47⁴³ + 43⁴³)
• D: None of these

Correct Answer: (b)

Solution: Simplify using modular arithmetic. Common factor = (47 + 43). The correct answer is (b).

Question 357: Find the product of all odd natural numbers less than 5000.

• A: 5000! / 2500 × 2501
• B: 5000! / 22500 × 2500!
• C: 5000! / 25000
• D: None of these

Correct Answer: (a)

Solution: Simplify using factorial properties. Product = 5000! / 2500 × 2501. The correct answer is (a).

Question 358: How many zeros will be required to number the pages of a book containing 1000 pages?

• A: 168
• B: 184
• C: 192
• D: 216

Correct Answer: (c)

Solution: Calculate zeros for each range (1-9, 10-99, etc.). Total zeros = 192. The correct answer is (c).

Question 359: If a² + b² + c² = 1, what is the maximum value of ab + bc + ca?

• A: 1/3
• B: 1
• C: 3√3
• D: 1

Correct Answer: (a)

Solution: Use inequality principles. Maximum value = 1/3. The correct answer is (a).

Question 360: Find the unit's digit in the sum of the fifth powers of the first 100 natural numbers.

• A: 0
• B: 2
• C: 5
• D: 8

Correct Answer: (a)

Solution: Simplify using modular arithmetic. Unit's digit = 0. The correct answer is (a).

Question 361: If the symbol [x] denotes the greatest integer less than or equal to x, then the value of [4/4] + [4/50] + [4/50] is

• A: 0
• B: 9
• C: 12
• D: 49

Correct Answer: (c)

Solution: Simplify step-by-step. Value = 12. The correct answer is (c).

Question 362: When 100²⁵ - 25 is written in decimal notation, the sum of its digits is

• A: 444
• B: 445
• C: 446
• D: 448

Correct Answer: (b)

Solution: Simplify using modular arithmetic. Sum of digits = 445. The correct answer is (b).

Question 363: What is the number of digits in the number (10²⁴)⁴

• A: 35
• B: 36
• C: 37
• D: 38

Correct Answer: (b)

Solution: Simplify using logarithms. Number of digits = 36. The correct answer is (b).

Question 364: Solve (0.07³ - 0.05³) / (0.07² + 0.07 × 0.05 + 0.05²)

• A: 0.002
• B: 0.02
• C: 0.2
• D: 0.0002

Correct Answer: (a)

Solution: Simplify using algebraic identity. Result = 0.002. The correct answer is (a).

Question 365: Which is not a prime number?

• A: 13
• B: 19
• C: 21
• D: 17

Correct Answer: (c)

Solution: Check divisibility. 21 is not a prime number. The correct answer is (c).

Question 366: If x = a(b - c), y = b(c - a), z = c(a - b), then the value of (x/a)³ + (y/b)³ + (z/c)³ is

• A: 2xyz / abc
• B: 3xyz / abc
• C: 0
• D: None of these

Correct Answer: (c)

Solution: Simplify using substitution. Result = 0. The correct answer is (c).

Question 367: Among the following statements, the statement which is not correct is:

• A: Every natural number is an integer.
• B: Every natural number is a real number.
• C: Every real number is a rational number.
• D: Every integer is a rational number.

Correct Answer: (c)

Solution: Real numbers include irrational numbers. The correct answer is (c).

Question 368: If a + b + c = 6 and ab + bc + ca = 10, then the value of a³ + b³ + c³ - 3abc is

• A: 36
• B: 48
• C: 42
• D: 40

Correct Answer: (a)

Solution: Use identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca). Result = 36. The correct answer is (a).

Question 369: If a - 1/a = 2, then the value of a³ - 1/a³ is

• A: 2
• B: 14
• C: 11
• D: 9

Correct Answer: (b)

Solution: Use identity a³ - 1/a³ = (a - 1/a)³ + 3(a - 1/a). Result = 14. The correct answer is (b).

Question 370: What is the value of a² + b² + c?

• A: Statement I alone is sufficient.
• B: Statement II alone is sufficient.
• C: Either statement alone is sufficient.
• D: Both statements together are necessary.

Correct Answer: (e)

Solution: Analyze both statements. Both are necessary. The correct answer is (e).

Question 371: The sum of digits of a two-digit number is 12 and the difference between the two digits is 6. What is the two-digit number?

• A: 39
• B: 84
• C: 93
• D: Other than the given options

Correct Answer: (c)

Solution: Solve using equations. Number = 93. The correct answer is (c).

Question 372: The difference between the greatest and the least four-digit numbers that begin with 3 and end with 5 is

• A: 900
• B: 909
• C: 990
• D: 999

Correct Answer: (c)

Solution: Greatest = 3995, Least = 3005. Difference = 990. The correct answer is (c).

Question 373: The sum of the perfect squares between 120 and 300 is

• A: 1204
• B: 1024
• C: 1296
• D: 1400

Correct Answer: (c)

Solution: List perfect squares: 121, 144, ..., 289. Sum = 1296. The correct answer is (c).

Question 374: If p³ - q³ = (p - q)(p - q)² - xpq, then find the value of x

• A: 1
• B: -3
• C: 3
• D: -1

Correct Answer: (d)

Solution: Expand and compare terms. x = -1. The correct answer is (d).

Question 375: What minimum value should be assigned to *, so that 2361*48 is exactly divisible by 9?

• A: 3
• B: 4
• C: 2
• D: 1

Correct Answer: (a)

Solution: Sum of digits must be divisible by 9. Minimum value = 3. The correct answer is (a).

Question 376: A, B, C, and D completed coloring a picture in 3/4 hour, 5/6 hour, 7/12 hour, and 7/6 hour respectively. Which took the least time?

• A: A
• B: B
• C: C
• D: D

Correct Answer: (c)

Solution: Compare times. Least time = 7/12 hours. The correct answer is (c).

Question 377: The difference between 4/5 of a number and 45% of the number is 56. What is 65% of the number?

• A: 96
• B: 104
• C: 112
• D: 120

Correct Answer: (b)

Solution: Solve using equations. 65% of number = 104. The correct answer is (b).

Question 378: If x + y : y + z : z + x = 6 : 7 : 8 and x + y + z = 14, find z

• A: 6
• B: 7
• C: 8
• D: 10

Correct Answer: (b)

Solution: Solve using ratios. z = 7. The correct answer is (b).

Question 379: One megabyte is

• A: 1048576 Bytes
• B: 1068576 Bytes
• C: 1058576 Bytes
• D: None of these

Correct Answer: (a)

Solution: Standard definition. 1 MB = 1048576 Bytes. The correct answer is (a).

Question 380: The number of three-digit numbers which are multiples of 9 is

• A: 100
• B: 99
• C: 98
• D: 101

Correct Answer: (a)

Solution: Count multiples of 9 from 108 to 999. Total = 100. The correct answer is (a).

Question 381: Two consecutive even positive integers, sum of the squares of which is 1060, are

• A: 12 and 14
• B: 20 and 22
• C: 22 and 24
• D: 15 and 18

Correct Answer: (c)

Solution: Solve using equations. Numbers = 22 and 24. The correct answer is (c).

Question 382: What is the number of trees planted in the field in rows and columns?

• A: Statement I alone is sufficient.
• B: Statement II alone is sufficient.
• C: Either statement alone is sufficient.
• D: Both statements together are necessary.

Correct Answer: (e)

Solution: Analyze both statements. Both are necessary. The correct answer is (e).

Question 383: If n is a natural number and n = p₁ˣ¹ × p₂ˣ² × p₃ˣ³, where p₁, p₂, p₃ are distinct prime factors, then the number of prime factors for n is

• A: x₁ + x₂ + x₃
• B: (x₁ + 1)(x₂ + 1)(x₃ + 1)
• C: x₁ × x₂ × x₃
• D: None of these

Correct Answer: (a)

Solution: The total number of prime factors is the sum of their exponents. Result = x₁ + x₂ + x₃. The correct answer is (a).

Question 384: Consider the following statements for the sequence of numbers given below: 11, 111, 1111, 11111, ... 1. Each number can be expressed in the form (4m + 3), where m is a natural number. 2. Some numbers are squares. Which of the above statements is/are correct?

• A: 1 only
• B: 2 only
• C: Both 1 and 2
• D: Neither 1 nor 2

Correct Answer: (a)

Solution: Statement 1 is true as all numbers fit the form (4m + 3). Statement 2 is false as none of the numbers are perfect squares. The correct answer is (a).

Question 385: An officer was appointed on maximum daily wages on contract money of ₹4956. But on being absent for some days, he was paid ₹3894. For how many days was he absent?

• A: 3
• B: 4
• C: 2
• D: None of these

Correct Answer: (a)

Solution: Calculate the difference in payment: ₹4956 - ₹3894 = ₹1062. Divide by daily wage: ₹1062 ÷ ₹354 = 3 days. The correct answer is (a).

Question 386: If the difference of the squares of two natural numbers is 19, find the sum of the squares of these numbers.

• A: 189
• B: 190
• C: 181
• D: None of these

Correct Answer: (b)

Solution: Let the numbers be x and y. Given x² - y² = 19 ⇒ (x - y)(x + y) = 19. Solve to get x = 10, y = 9. Sum of squares = 10² + 9² = 190. The correct answer is (b).

Question 387: If the sum of two numbers is 14 and their difference is 10, find the product of these two numbers.

• A: 18
• B: 20
• C: 24
• D: 22

Correct Answer: (c)

Solution: Let the numbers be x and y. Solve equations: x + y = 14, x - y = 10. Get x = 12, y = 2. Product = 12 × 2 = 24. The correct answer is (c).

Question 388: What should be added to 2x² + 3x - 5 to make it x² + x + 1?

• A: -x² - 4x + 6
• B: x² - 4x + 6
• C: x² - 4x + 6
• D: x² - 4x + 6

Correct Answer: (a)

Solution: Subtract (2x² + 3x - 5) from (x² + x + 1): (x² + x + 1) - (2x² + 3x - 5) = -x² - 4x + 6. The correct answer is (a).