← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 617 · 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.What is the smallest number you must add to 98 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
  2. 2.What is the smallest number you must add to 49 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  3. 3.Which of these numbers is divisible by 6? (a) 1412 (b) 1406 (c) 1418 (d) 1410
  4. 4.Which of these numbers is divisible by 8? (a) 860 (b) 848 (c) 846 (d) 849
  5. 5.What is the remainder when 239 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
  6. 6.Which of these numbers is divisible by 3? (a) 1637 (b) 1630 (c) 1632 (d) 1634
  7. 7.What is the units digit of 2^24? (a) 6 (b) 8 (c) 4 (d) 2
  8. 8.What is the units digit of 8^28? (a) 8 (b) 2 (c) 6 (d) 4
  9. 9.Which of these numbers is divisible by 4? (a) 2793 (b) 2797 (c) 2802 (d) 2796
  10. 10.What is the smallest number you must add to 269 to make it divisible by 12? (a) 6 (b) 5 (c) 7 (d) 8
  11. 11.What is the smallest number you must add to 626 to make it divisible by 9? (a) 5 (b) 3 (c) 4 (d) 6
  12. 12.What is the smallest number you must add to 602 to make it divisible by 9? (a) 8 (b) 2 (c) 3 (d) 1
  13. 13.What is the units digit of 4^20? (a) 4 (b) 7 (c) 5 (d) 6
  14. 14.What is the units digit of 2^34? (a) 6 (b) 4 (c) 8 (d) 2
  15. 15.Which of these numbers is divisible by 4? (a) 3301 (b) 3311 (c) 3305 (d) 3304
  16. 16.What is the remainder when 296 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
  17. 17.Which of these numbers is divisible by 8? (a) 5640 (b) 5635 (c) 5652 (d) 5641
  18. 18.What is the smallest number you must add to 310 to make it divisible by 8? (a) 1 (b) 3 (c) 6 (d) 2
  19. 19.What is the smallest number you must add to 896 to make it divisible by 12? (a) 5 (b) 8 (c) 3 (d) 4
  20. 20.What is the units digit of 2^50? (a) 6 (b) 8 (c) 2 (d) 4
  21. 21.What is the smallest number you must add to 520 to make it divisible by 12? (a) 9 (b) 4 (c) 8 (d) 7
  22. 22.Which of these numbers is divisible by 8? (a) 2789 (b) 2803 (c) 2794 (d) 2792
  23. 23.What is the units digit of 2^55? (a) 2 (b) 8 (c) 6 (d) 4
  24. 24.What is the remainder when 659 is divided by 10? (a) 8 (b) 1 (c) 9 (d) 7
  25. 25.What is the remainder when 445 is divided by 6? (a) 1 (b) 0 (c) 5 (d) 2
  26. 26.What is the units digit of 2^11? (a) 6 (b) 4 (c) 8 (d) 2
  27. 27.What is the units digit of 4^54? (a) 5 (b) 7 (c) 4 (d) 6
  28. 28.What is the remainder when 252 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
  29. 29.What is the remainder when 402 is divided by 9? (a) 5 (b) 3 (c) 7 (d) 6
  30. 30.What is the smallest number you must add to 91 to make it divisible by 3? (a) 1 (b) 2
  31. 31.What is the units digit of 3^24? (a) 7 (b) 1 (c) 9 (d) 3
  32. 32.What is the remainder when 157 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
  33. 33.What is the smallest number you must add to 184 to make it divisible by 10? (a) 4 (b) 6 (c) 5 (d) 7
  34. 34.What is the remainder when 322 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
  35. 35.What is the remainder when 106 is divided by 3? (a) 1 (b) 0 (c) 2
  36. 36.What is the units digit of 2^54? (a) 6 (b) 2 (c) 4 (d) 8
  37. 37.What is the smallest number you must add to 440 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
  38. 38.What is the smallest number you must add to 63 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  39. 39.What is the smallest number you must add to 122 to make it divisible by 9? (a) 3 (b) 5 (c) 4 (d) 6
  40. 40.Which of these numbers is divisible by 8? (a) 2642 (b) 2657 (c) 2648 (d) 2654

Answer key

  1. 2
  2. 3
  3. 1410
  4. 848
  5. 4
  6. 1632
  7. 6
  8. 6
  9. 2796
  10. 7
  11. 4
  12. 1
  13. 6
  14. 4
  15. 3304
  16. 1
  17. 5640
  18. 2
  19. 4
  20. 4
  21. 8
  22. 2792
  23. 8
  24. 9
  25. 1
  26. 8
  27. 6
  28. 2
  29. 6
  30. 2
  31. 1
  32. 2
  33. 6
  34. 2
  35. 1
  36. 4
  37. 4
  38. 1
  39. 4
  40. 2648

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