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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 949 · 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 551 to make it divisible by 10? (a) 7 (b) 1 (c) 8 (d) 9
  2. 2.What is the remainder when 604 is divided by 8? (a) 5 (b) 6 (c) 3 (d) 4
  3. 3.What is the smallest number you must add to 657 to make it divisible by 10? (a) 2 (b) 7 (c) 4 (d) 3
  4. 4.What is the smallest number you must add to 277 to make it divisible by 12? (a) 10 (b) 1 (c) 9 (d) 11
  5. 5.What is the units digit of 9^16? (a) 1 (b) 2 (c) 9 (d) 0
  6. 6.What is the units digit of 4^50? (a) 5 (b) 4 (c) 7 (d) 6
  7. 7.What is the smallest number you must add to 121 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  8. 8.What is the remainder when 387 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
  9. 9.What is the remainder when 245 is divided by 10? (a) 7 (b) 4 (c) 5 (d) 6
  10. 10.Which of these numbers is divisible by 9? (a) 5627 (b) 5634 (c) 5639 (d) 5646
  11. 11.What is the smallest number you must add to 364 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
  12. 12.What is the units digit of 4^55? (a) 4 (b) 5 (c) 3 (d) 6
  13. 13.What is the units digit of 4^23? (a) 5 (b) 6 (c) 3 (d) 4
  14. 14.Which of these numbers is divisible by 4? (a) 494 (b) 491 (c) 485 (d) 488
  15. 15.What is the smallest number you must add to 209 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  16. 16.What is the units digit of 9^45? (a) 9 (b) 8 (c) 7 (d) 1
  17. 17.What is the smallest number you must add to 371 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  18. 18.What is the smallest number you must add to 236 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  19. 19.What is the units digit of 3^20? (a) 7 (b) 1 (c) 3 (d) 9
  20. 20.What is the units digit of 3^17? (a) 7 (b) 1 (c) 3 (d) 9
  21. 21.What is the smallest number you must add to 599 to make it divisible by 9? (a) 4 (b) 3 (c) 5 (d) 6
  22. 22.What is the remainder when 833 is divided by 11? (a) 3 (b) 8 (c) 7 (d) 9
  23. 23.What is the remainder when 129 is divided by 10? (a) 9 (b) 1 (c) 7 (d) 8
  24. 24.What is the smallest number you must add to 49 to make it divisible by 3? (a) 1 (b) 2
  25. 25.What is the remainder when 213 is divided by 9? (a) 7 (b) 6 (c) 3 (d) 5
  26. 26.What is the units digit of 8^22? (a) 4 (b) 8 (c) 6 (d) 2
  27. 27.What is the smallest number you must add to 147 to make it divisible by 8? (a) 3 (b) 5 (c) 6 (d) 4
  28. 28.What is the remainder when 290 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 2
  29. 29.Which of these numbers is divisible by 8? (a) 1593 (b) 1580 (c) 1584 (d) 1586
  30. 30.What is the units digit of 2^14? (a) 2 (b) 4 (c) 8 (d) 6
  31. 31.What is the units digit of 9^12? (a) 9 (b) 1 (c) 0 (d) 2
  32. 32.What is the remainder when 116 is divided by 3? (a) 1 (b) 2 (c) 0
  33. 33.Which of these numbers is divisible by 3? (a) 2637 (b) 2638 (c) 2635 (d) 2641
  34. 34.What is the units digit of 2^40? (a) 4 (b) 8 (c) 6 (d) 2
  35. 35.What is the units digit of 9^19? (a) 8 (b) 1 (c) 7 (d) 9
  36. 36.What is the smallest number you must add to 704 to make it divisible by 9? (a) 7 (b) 2 (c) 8 (d) 6
  37. 37.What is the remainder when 391 is divided by 7? (a) 5 (b) 1 (c) 6 (d) 4
  38. 38.What is the smallest number you must add to 358 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 6
  39. 39.What is the smallest number you must add to 531 to make it divisible by 7? (a) 1 (b) 6 (c) 2 (d) 3
  40. 40.What is the smallest number you must add to 929 to make it divisible by 12? (a) 5 (b) 8 (c) 6 (d) 7

Answer key

  1. 9
  2. 4
  3. 3
  4. 11
  5. 1
  6. 6
  7. 3
  8. 2
  9. 5
  10. 5634
  11. 2
  12. 4
  13. 4
  14. 488
  15. 3
  16. 9
  17. 4
  18. 4
  19. 1
  20. 3
  21. 4
  22. 8
  23. 9
  24. 2
  25. 6
  26. 4
  27. 5
  28. 2
  29. 1584
  30. 4
  31. 1
  32. 2
  33. 2637
  34. 6
  35. 9
  36. 7
  37. 6
  38. 2
  39. 1
  40. 7

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