← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 851 · 40 questions
TalentJR

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

NameDateTime takenScore ___ / 40
  1. 1.Which of these numbers is divisible by 3? (a) 2377 (b) 2372 (c) 2373 (d) 2374
  2. 2.Which of these numbers is divisible by 4? (a) 616 (b) 614 (c) 622 (d) 619
  3. 3.Which of these numbers is divisible by 4? (a) 3148 (b) 3155 (c) 3151 (d) 3147
  4. 4.What is the remainder when 290 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
  5. 5.Which of these numbers is divisible by 3? (a) 737 (b) 735 (c) 740 (d) 734
  6. 6.What is the units digit of 2^27? (a) 8 (b) 6 (c) 2 (d) 4
  7. 7.What is the smallest number you must add to 475 to make it divisible by 8? (a) 6 (b) 4 (c) 5 (d) 3
  8. 8.What is the units digit of 8^36? (a) 8 (b) 6 (c) 2 (d) 4
  9. 9.What is the smallest number you must add to 242 to make it divisible by 3? (a) 1 (b) 2
  10. 10.What is the units digit of 2^14? (a) 8 (b) 4 (c) 6 (d) 2
  11. 11.What is the smallest number you must add to 444 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 7
  12. 12.Which of these numbers is divisible by 4? (a) 2699 (b) 2706 (c) 2701 (d) 2700
  13. 13.What is the units digit of 2^18? (a) 4 (b) 8 (c) 6 (d) 2
  14. 14.What is the smallest number you must add to 674 to make it divisible by 9? (a) 3 (b) 1 (c) 2 (d) 8
  15. 15.What is the remainder when 512 is divided by 9? (a) 6 (b) 1 (c) 7 (d) 8
  16. 16.What is the smallest number you must add to 107 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 1
  17. 17.What is the units digit of 4^41? (a) 5 (b) 3 (c) 6 (d) 4
  18. 18.What is the remainder when 424 is divided by 7? (a) 5 (b) 3 (c) 4 (d) 6
  19. 19.Which of these numbers is divisible by 9? (a) 404 (b) 406 (c) 422 (d) 405
  20. 20.Which of these numbers is divisible by 8? (a) 1416 (b) 1411 (c) 1417 (d) 1428
  21. 21.Which of these numbers is divisible by 8? (a) 7032 (b) 7033 (c) 7042 (d) 7028
  22. 22.What is the remainder when 359 is divided by 10? (a) 8 (b) 1 (c) 9 (d) 7
  23. 23.Which of these numbers is divisible by 3? (a) 1097 (b) 1103 (c) 1099 (d) 1098
  24. 24.What is the smallest number you must add to 473 to make it divisible by 10? (a) 8 (b) 3 (c) 6 (d) 7
  25. 25.Which of these numbers is divisible by 4? (a) 200 (b) 207 (c) 198 (d) 202
  26. 26.What is the smallest number you must add to 397 to make it divisible by 9? (a) 6 (b) 7 (c) 1 (d) 8
  27. 27.What is the units digit of 2^17? (a) 2 (b) 4 (c) 8 (d) 6
  28. 28.What is the smallest number you must add to 262 to make it divisible by 6? (a) 4 (b) 2 (c) 1 (d) 3
  29. 29.What is the remainder when 368 is divided by 7? (a) 5 (b) 4 (c) 3 (d) 6
  30. 30.What is the smallest number you must add to 142 to make it divisible by 8? (a) 6 (b) 3 (c) 2 (d) 1
  31. 31.What is the remainder when 831 is divided by 12? (a) 2 (b) 4 (c) 3 (d) 9
  32. 32.Which of these numbers is divisible by 9? (a) 4468 (b) 4459 (c) 4455 (d) 4451
  33. 33.What is the remainder when 314 is divided by 9? (a) 7 (b) 6 (c) 8 (d) 1
  34. 34.What is the units digit of 8^47? (a) 8 (b) 4 (c) 6 (d) 2
  35. 35.What is the units digit of 2^34? (a) 8 (b) 2 (c) 6 (d) 4
  36. 36.What is the units digit of 4^32? (a) 5 (b) 6 (c) 4 (d) 7
  37. 37.Which of these numbers is divisible by 6? (a) 2509 (b) 2507 (c) 2518 (d) 2508
  38. 38.What is the smallest number you must add to 432 to make it divisible by 10? (a) 8 (b) 7 (c) 9 (d) 2
  39. 39.What is the remainder when 678 is divided by 12? (a) 5 (b) 8 (c) 6 (d) 7
  40. 40.What is the smallest number you must add to 636 to make it divisible by 8? (a) 6 (b) 5 (c) 3 (d) 4

Answer key

  1. 2373
  2. 616
  3. 3148
  4. 2
  5. 735
  6. 8
  7. 5
  8. 6
  9. 1
  10. 4
  11. 6
  12. 2700
  13. 4
  14. 1
  15. 8
  16. 1
  17. 4
  18. 4
  19. 405
  20. 1416
  21. 7032
  22. 9
  23. 1098
  24. 7
  25. 200
  26. 8
  27. 2
  28. 2
  29. 4
  30. 2
  31. 3
  32. 4455
  33. 8
  34. 2
  35. 4
  36. 6
  37. 2508
  38. 8
  39. 6
  40. 4

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad