Practice Number Theory MCQs for Bank of Punjab (BoP) Officer / OG-II Mathematics — topic-wise sets with solved answers.
Q1. There are four prime numbers written in ascending order of magnitude. The product of the first three is 7429 and of the last three is 12673. Find the fourth number?
Answer: 29
Explanation: 7429 = 17×19×23 and 12673 = 19×23×29; the fourth (largest) number is 29.
Q2. Find the greatest number of four digits which when divided by 10, 15, 21 and 28 leaves 4, 9, 15 and 22 as remainders respectively?
Answer: 9654
Explanation: Each divisor exceeds its remainder by 6; LCM(10,15,21,28) = 420; largest 4-digit multiple = 9660; answer = 9660 − 6 = 9654.
Q3. What is the least number which when divided by 8, 12, 18 and 24 leaves the remainders 4, 8, 14 and 20 respectively?
Answer: 68
Explanation: Each remainder is 4 less than its divisor, so the answer = LCM(8,12,18,24)−4 = 72−4 = 68.
Q4. LCM of 87 and 145 is_______?
Answer: 435
Explanation: 87 = 3×29, 145 = 5×29; LCM = 3×5×29 = 435. Show the calculation clearly when solving similar quantitative items.
Q5. Find the greatest number which will divide 25, 73 and 97 as so to leave the same remainder in each case?
Answer: 24
Explanation: The required number is the GCD of the differences: GCD(73−25, 97−73) = GCD(48,24) = 24.
Q6. Which of the following numbers is divisible by 2, 3, and 5?
Answer: 390
Explanation: 390 is divisible by 2 (even), by 3 (3+9+0=12), and by 5 (ends in 0); 575 and 675 fail divisibility by 2, and 590 fails divisibility by 3.
Q7. Find the least number which when divided by 6, 7, 8, 9 and 10 leaves 1, 2, 3, 4 and 5 as remainders respectively, but when divided by 19 leaves no remainder?
Answer: 5035
Explanation: LCM(6,7,8,9,10) = 2520; each remainder is exactly 5 less than its divisor, so the number is of the form 2520k − 5. Testing k=2 gives 5035, which is divisible by 19 (5035 ÷ 19 = 265).
Q8. The least number of five digits that have 144 their HCF is__________?
Answer: 10080
Explanation: The LCM of 144 is 144 itself; the smallest five-digit multiple of 144 is 144 × 70 = 10080.
Q9. A man donated 5 % of his income to a charitable organization and deposited 20 % of the remainder in a bank. If he now has Rs. 1919 left, what is his income?
Answer: Rs. 2525
Explanation: After donating 5%, remainder = 0.95x; after depositing 20%, left with 0.8 × 0.95x = 0.76x = 1919. Solving gives x = Rs. 2525.
Q10. Find the greatest 4-digit number exactly divisible by 3, 4 and 5?
Answer: 9960
Explanation: LCM(3,4,5) = 60; the greatest 4-digit multiple of 60 is 9960 (since 9999÷60 = 166.65, so 166×60 = 9960).
Q11. A merchant has three different types of milk: 435 liters, 493 liters and 551 liters. Find the least number of casks of equal size required to store all the milk without mixing. HCF
Answer: 51
Explanation: HCF(435, 493, 551) = 29 (since differences 58 = 2×29 and HCF(435,29) = 29). Number of casks = 435/29 + 493/29 + 551/29 = 15 + 17 + 19 = 51.
Q12. Greatest Common Divisor of two numbers is 8 while their Least Common Multiple is 144. Find the other number if one number is 16. HCF
Answer: 72
Explanation: Using HCF × LCM = product of two numbers: 8 × 144 = 16 × x, so x = 72.
Q13. What will be the least number which when doubled will be exactly divisible by 12, 18, 21 and 30?
Answer: 630
Explanation: LCM(12,18,21,30) = 1260; the least number that when doubled equals 1260 is 630.
Q14. Least perfect square number, exactly divisible by 21, 36 and 56 is_________?
Answer: 7056
Explanation: LCM(21, 36, 56) = 504 = 2³ × 3² × 7; to make it a perfect square, multiply by 2 × 7 = 14, giving 504 × 14 = 7056.
Q15. A, B and C start at the same time in the same direction to run around a circular stadium.
Answer: 46 min 12 sec
Explanation: LCM(252, 308, 198) = 2772 seconds = 46 minutes 12 seconds, at which point all three runners meet at the starting point again.
Q16. Find the least number which when divided by 35 leaves remainder 25, when divided by 25 leaves remainder 15 and when divided by 15 leaves remainder 5?
Answer: 515
Explanation: Each remainder is 10 less than its divisor, so the number = LCM(35,25,15) − 10 = 525 − 10 = 515.
Q17. The least number which when divided by 16, 18 and 21, leave the remainder 3, 5 and 8 respectively is___________?
Answer: 995
Explanation: LCM(16,18,21) = 1008. Each remainder is 13 less than its divisor (16−3=13, 18−5=13, 21−8=13), so the number = 1008 − 13 = 995.
Q18. The product of two numbers is 2025 and their HCF is 15 their LCM is____________?
Answer: 135
Explanation: For any two numbers: LCM × HCF = product of the numbers. LCM = 2025/15 = 135.
Q19. The greatest number of four digits which is divisible by 15, 25, 40 and 75 is_________?
Answer: 9600
Explanation: LCM(15, 25, 40, 75) = 600; the greatest four-digit multiple of 600 is 600 × 16 = 9600.
Q20. LCM of 1/3, 5/6, 5/4, 10/7 is_________?
Answer: 10
Explanation: LCM of fractions = LCM of numerators / GCD of denominators = LCM(1,5,5,10) / GCD(3,6,4,7) = 10/1 = 10.
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