Q1: 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).
Q2: 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).
Q3: 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).
Q4: 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).
Q5: 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).
Q6: 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).
Q7: 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).
Q8: 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).
Q9: 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).
Q10: 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).
Q11: 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).
Q12: 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).
Q13: 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).
Q14: √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).
Q15: 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).
Q16: 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).
Q17: 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).
Q18: 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).
Q19: 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).
Q20: 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).
Q21: 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).
Q22: 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).
Q23: 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).
Q24: 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).
Q25: 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).
Q26: 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).
Q27: 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).
Q28: 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).
Q29: 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).
Q30: 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).
Q31: 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).
Q32: 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).
Q33: 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).
Q34: 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).
Q35: 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).
Q36: 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).
Q37: 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).
Q38: 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).
Q39: 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).
Q40: 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).
Q41: 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).
Q42: 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).
Q43: 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).
Q44: 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).
Q45: 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).
Q46: 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).
Q47: 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).
Q48: 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).
Q49: 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).
Q50: 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).
Q51: 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).
Q52: 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).
Q53: 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).
Q54: 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).
Q55: 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).
Q56: 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).
Q57: 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).
Q58: 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).
Q59: 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).
Q60: 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).
Q61: 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).
Q62: 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).
Q63: 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).
Q64: 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).
Q65: 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).
Q66: 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).
Q67: 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).
Q68: 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).
Q69: 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).
Q70: 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).
Q71: 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).
Q72: 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).
Q73: 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).
Q74: 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).
Q75: 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).
Q76: 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).
Q77: 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).
Q78: 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).
Q79: 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).
Q80: 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).
Q81: 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).
Q82: 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).
Q83: 14 ÷ 9 = ?
A. 9
B. 10
C. 81
D. 810
Correct Answer: (a)
Solution: 14 ÷ 9 = 1 remainder 5. The correct answer is (a).
Q84: 136 × 12 × 8 = ?
A. 12066
B. 13046
C. 13064
D. 13066
Correct Answer: (d)
Solution: 136 × 12 × 8 = 13066. The correct answer is (d).
Q85: 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).
Q86: 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).
Q87: 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).
Q88: (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).
Q89: 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).
Q90: Multiply 5746320819 by 125
A. 718290102375
B. 728490301375
C. 748290103375
D. 798290102975
Correct Answer: (a)
Solution: 5746320819 × 125 = 718290102375. The correct answer is (a).
Q91: 935421 × 625 = ?
A. 575648125
B. 584638125
C. 584649125
D. 585628125
Correct Answer: (c)
Solution: 935421 × 625 = 584649125. The correct answer is (c).
Q92: (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).
Q93: (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).
Q94: (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).
Q95: 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).
Q96: 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).
Q97: 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).
Q98: 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).
Q99: (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).
Q100: -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).
Q101: 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).
Q102: 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).
Q103: 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).
Q104: 1256 × 3892 = ?
A. 4883852
B. 4888532
C. 4888352
D. 4883582
Correct Answer: (c)
Solution: Direct multiplication gives 4888352. The correct answer is (c).
Q105: 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).
Q106: 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).
Q107: (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).
Q108: 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).
Q109: 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).
Q110: 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).
Q111: {(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).
Q112: 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).
Q113: 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).
Q114: 935421 × 625 = ?
A. 575648125
B. 584638125
C. 584649125
D. 585628125
Correct Answer: (c)
Solution: Multiplying: 935421 × 625 = 584649125. The correct answer is (c).
Q115: (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).
Q116: (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).
Q117: (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).
Q118: 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).
Q119: 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).
Q120: 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).
Q121: 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).
Q122: (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).
Q123: 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).
Q124: 60840 ÷ 234 = ?
A. 225
B. 255
C. 260
D. 310
Correct Answer: (c)
Solution: Dividing: 60840 ÷ 234 = 260. The correct answer is (c).
Q125: (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).
Q126: 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).
Q127: 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).
Q128: 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).
Q129: 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).
Q130: 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).
Q131: 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).
Q132: 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).
Q133: 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).
Q134: 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).
Q135: (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).
Q136: 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).
Q137: 60840 ÷ 234 = ?
A. 225
B. 255
C. 260
D. 310
Correct Answer: (c)
Solution: Dividing: 60840 ÷ 234 = 260. The correct answer is (c).
Q138: (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).
Q139: (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).
Q140: 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).
Q141: 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).
Q142: 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).
Q143: 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).
Q144: 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).
Q145: 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).
Q146: 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).
Q147: 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).
Q148: √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).
Q149: 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).
Q150: 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).
Q151: 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).
Q152: 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).
Q153: 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).
Q154: 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).
Q155: 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).
Q156: 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).
Q157: 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).
Q158: 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).
Q159: 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).
Q160: 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).
Q161: 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).
Q162: 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
Q163: 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).
Q164: 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).
Q165: 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).
Q166: 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).
Q167: 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).
Q168: 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).
Q169: 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).
Q170: 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).
Q171: 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).
Q172: 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).
Q173: 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).
Q174: 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).
Q175: 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).
Q176: 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).
Q177: 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).
Q178: 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).
Q179: 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).
Q180: 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).
Q181: 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).
Q182: 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).
Q183: 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).
Q184: 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).
Q185: 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).
Q186: 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).
Q187: 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).
Q188: 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).
Q189: 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).
Q190: 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).
Q191: 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).
Q192: 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).
Q193: 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).
Q194: 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).
Q195: 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).
Q196: 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).
Q197: 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).
Q198: 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).
Q199: 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).
Q200: 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).
Q201: 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).
Q202: 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).
Q203: 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).
Q204: 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).
Q205: 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).
Q206: 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).
Q207: 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).
Q208: 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).
Q209: 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).
Q210: 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).
Q211: 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).
Q212: 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).
Q213: 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).
Q214: 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).
Q215: 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).
Q216: 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).
Q217: 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).
Q218: 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).
Q219: 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).
Q220: 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).
Q221: 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).
Q222: 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).
Q223: 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).
Q224: 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).
Q225: 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).
Q226: 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).
Q227: 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).
Q228: 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).
Q229: 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).
Q230: 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).
Q231: 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).
Q232: 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).
Q233: 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).
Q234: 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).
Q235: 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).
Q236: 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).
Q237: 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).
Q238: 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).
Q239: 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).
Q240: 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).
Q241: 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).
Q242: 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).
Q243: 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).
Q244: 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).
Q245: 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).
Q246: 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).
Q247: 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).
Q248: 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).
Q249: 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).
Q250: 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).
Q251: 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).
Q252: 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).
Q253: 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).
Q254: 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).
Q255: 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).
Q256: 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).
Q257: 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).
Q258: 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).
Q259: (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).
Q260: 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).
Q261: 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).
Q262: 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).
Q263: 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).
Q264: 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).
Q265: 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).
Q266: 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).
Q267: 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).
Q268: 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).
Q269: 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).
Q270: 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).
Q271: 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).
Q272: 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).
Q273: 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).
Q274: 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).
Q275: 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).
Q276: 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).
Q277: 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).
Q278: 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).
Q279: 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).
Q280: 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).
Q281: 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).
Q282: 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).
Q283: 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).
Q284: 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).
Q285: 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).
Q286: (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).
Q287: 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).
Q288: (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).
Q289: (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).
Q290: (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).
Q291: 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).
Q292: 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).
Q293: 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).
Q294: 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).
Q295: 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).
Q296: 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).
Q297: 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).
Q298: 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).
Q299: 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).
Q300: 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).
Q301: 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).
Q302: 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).
Q303: 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).
Q304: 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).
Q305: 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).
Q306: 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).
Q307: 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).
Q308: 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).
Q309: 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).
Q310: 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).
Q311: 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).
Q312: 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).
Q313: 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).
Q314: 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).
Q315: 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).
Q316: 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).
Q317: 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).
Q318: 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).
Q319: 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).
Q320: 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).
Q321: 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).
Q322: 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).
Q323: 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).
Q324: (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).
Q325: 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).
Q326: (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).
Q327: (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).
Q328: (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).
Q329: 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).
Q330: 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).
Q331: 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).
Q332: 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).
Q333: 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).
Q334: 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).
Q335: 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).
Q336: 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).
Q337: 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).
Q338: 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).
Q339: 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).
Q340: 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).
Q341: 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).
Q342: 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).
Q343: 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).
Q344: 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).
Q345: 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).
Q346: 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).
Q347: 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).
Q348: 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).
Q349: 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).
Q350: 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).
Q351: 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).
Q352: 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).
Q353: 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).
Q354: 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).
Q355: 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).
Q356: 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).
Q357: 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).
Q358: 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).
Q359: 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).
Q360: 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).
Q361: 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).
Q362: 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).
Q363: 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).
Q364: 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).
Q365: 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).
Q366: 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).
Q367: 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).
Q368: 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).
Q369: 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).
Q370: 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).
Q371: 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).
Q372: 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).
Q373: 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).
Q374: 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).
Q375: 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).
Q376: 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).
Q377: 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).
Q378: 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).
Q379: 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).
Q380: 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).
