14. Practice Question Set B — Moderate / Exam Level (30 MCQs)

Q1. A number is formed as 1 x 2 5 3 6 (a 6-digit number). If this number is divisible by 72, find the value of x. (a) 0 (b) 1 (c) 2 (d) 3

Q2. What is the remainder when 4,753,296 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 4

Q3. Which of the following six-digit numbers is exactly divisible by 13: 6,214,357 or 1,001,000? (a) Only 6,214,357 (b) Only 1,001,000 (c) Both (d) Neither

Q4. How many trailing zeros are there in 125!? (a) 28 (b) 29 (c) 30 (d) 31

Q5. Find the highest power of 5 that divides 173! completely. (a) 39 (b) 40 (c) 41 (d) 42

Q6. Find the highest power of 3 that divides 50! completely. (a) 20 (b) 21 (c) 22 (d) 23

Q7. If n is odd, 5^n + 2^n is always divisible by: (a) 3 (b) 5 (c) 7 (d) 10

Q8. What is the remainder when 2^(96) is divided by 7? (a) 0 (b) 1 (c) 2 (d) 4

Q9. What is the smallest value of n such that n! ends in exactly 24 zeros? (a) 96 (b) 99 (c) 100 (d) 105

Q10. What least number must be added to 5,634 to make it exactly divisible by 7? (a) 1 (b) 2 (c) 3 (d) 4

Q11. In the five-digit number 4 8 2 1 3 x, the last three digits “13x” must make the number divisible by 8. Find the value of the units digit x. (a) 2 (b) 4 (c) 6 (d) 8

Q12. The product of any 4 consecutive positive integers is always divisible by: (a) 12 (b) 16 (c) 24 (d) 30

Q13. A number N = 63a5b is divisible by both 3 and 5. If b=0, how many possible values can a take (digits 0–9)? (a) 2 (b) 3 (c) 4 (d) 5

Q14. What is the remainder when 7^(84) is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4

Q15. A number when divided by 342 gives remainder 47. What will be the remainder when the same number is divided by 18? (Note: 342 = 18 × 19) (a) 5 (b) 7 (c) 9 (d) 11

Q16. A 6-digit number is written as 5 x 4 7 3 y. If it is exactly divisible by 72, find x+y. (a) 6 (b) 7 (c) 8 (d) 9

Q17. Using the 1001-grouping trick, what is the remainder when 8,462,573 is divided by 13? (a) 0 (b) 1 (c) 2 (d) 3

Q18. Find the highest power of 6 that divides 100! completely. (a) 33 (b) 48 (c) 50 (d) 97

Q19. What is the remainder when 30! is divided by 31? (a) 0 (b) 1 (c) 15 (d) 30

Q20. The product of any 6 consecutive positive integers is always divisible by: (a) 120 (b) 360 (c) 720 (d) 5040

Q21. Find the smallest number which, when divided by 12, 16, 18, or 30, always leaves a remainder of 7. (a) 720 (b) 727 (c) 730 (d) 737

Q22. How many numbers from 1 to 600 are divisible by 4 or 6? (a) 150 (b) 175 (c) 200 (d) 225

Q23. The digit sum of 987,654,321 is 45. What is the remainder when this number is divided by 9? (a) 0 (b) 3 (c) 6 (d) 9

Q24. 17^(15) + 13^(15) is always divisible by: (a) 7 (b) 11 (c) 13 (d) 30

Q25. What is the remainder when 3^(250) is divided by 7? (a) 1 (b) 2 (c) 4 (d) 6

Q26. Find x so that the seven-digit number 3 x 2 8 4 7 1 is divisible by 11. (a) 4 (b) 5 (c) 6 (d) 7

Q27. How many factors of 2,520 lie strictly between 10 and 100? (a) 18 (b) 19 (c) 20 (d) 21

Q28. Which of the following is divisible by 13: 4,58,730 or 4,58,731? (a) Only 4,58,730 (b) Only 4,58,731 (c) Both (d) Neither

Q29. What is the remainder when 2^(100) is divided by 9? (a) 1 (b) 4 (c) 7 (d) 8

Q30. Find the smallest number which, when divided by 8, 9, or 10, always leaves a remainder of 5. (a) 355 (b) 360 (c) 365 (d) 370