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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 699 · 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 units digit of 2^29? (a) 4 (b) 8 (c) 2 (d) 6
  2. 2.What is the remainder when 718 is divided by 10? (a) 2 (b) 9 (c) 8 (d) 7
  3. 3.What is the remainder when 608 is divided by 9? (a) 6 (b) 7 (c) 4 (d) 5
  4. 4.What is the smallest number you must add to 153 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 6
  5. 5.What is the units digit of 8^33? (a) 2 (b) 4 (c) 8 (d) 6
  6. 6.What is the remainder when 299 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
  7. 7.What is the smallest number you must add to 376 to make it divisible by 9? (a) 1 (b) 3 (c) 7 (d) 2
  8. 8.What is the remainder when 129 is divided by 8? (a) 1 (b) 7 (c) 0 (d) 2
  9. 9.What is the smallest number you must add to 733 to make it divisible by 11? (a) 3 (b) 4 (c) 7 (d) 5
  10. 10.What is the units digit of 2^12? (a) 4 (b) 2 (c) 8 (d) 6
  11. 11.What is the units digit of 9^28? (a) 9 (b) 1 (c) 2 (d) 0
  12. 12.What is the smallest number you must add to 136 to make it divisible by 3? (a) 1 (b) 2
  13. 13.Which of these numbers is divisible by 9? (a) 2511 (b) 2524 (c) 2505 (d) 2517
  14. 14.What is the remainder when 247 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
  15. 15.What is the remainder when 255 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
  16. 16.What is the remainder when 151 is divided by 5? (a) 1 (b) 4 (c) 0 (d) 2
  17. 17.What is the smallest number you must add to 743 to make it divisible by 12? (a) 11 (b) 3 (c) 2 (d) 1
  18. 18.What is the units digit of 3^20? (a) 7 (b) 3 (c) 9 (d) 1
  19. 19.Which of these numbers is divisible by 6? (a) 2939 (b) 2933 (c) 2941 (d) 2934
  20. 20.Which of these numbers is divisible by 4? (a) 2286 (b) 2279 (c) 2280 (d) 2282
  21. 21.What is the units digit of 3^19? (a) 7 (b) 3 (c) 9 (d) 1
  22. 22.What is the remainder when 91 is divided by 3? (a) 2 (b) 1 (c) 0
  23. 23.What is the remainder when 219 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
  24. 24.What is the smallest number you must add to 203 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  25. 25.What is the smallest number you must add to 242 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 2
  26. 26.What is the remainder when 394 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
  27. 27.What is the smallest number you must add to 86 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
  28. 28.What is the units digit of 8^10? (a) 8 (b) 4 (c) 6 (d) 2
  29. 29.What is the units digit of 7^57? (a) 9 (b) 7 (c) 1 (d) 3
  30. 30.Which of these numbers is divisible by 9? (a) 1575 (b) 1586 (c) 1581 (d) 1573
  31. 31.Which of these numbers is divisible by 9? (a) 1684 (b) 1695 (c) 1702 (d) 1692
  32. 32.Which of these numbers is divisible by 3? (a) 1376 (b) 1378 (c) 1377 (d) 1381
  33. 33.What is the smallest number you must add to 161 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
  34. 34.What is the remainder when 502 is divided by 10? (a) 1 (b) 2 (c) 8 (d) 3
  35. 35.What is the remainder when 306 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
  36. 36.What is the units digit of 2^50? (a) 6 (b) 8 (c) 4 (d) 2
  37. 37.What is the smallest number you must add to 157 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  38. 38.What is the units digit of 7^38? (a) 9 (b) 3 (c) 7 (d) 1
  39. 39.What is the smallest number you must add to 331 to make it divisible by 10? (a) 7 (b) 9 (c) 8 (d) 1
  40. 40.What is the remainder when 647 is divided by 9? (a) 6 (b) 8 (c) 1 (d) 7

Answer key

  1. 2
  2. 8
  3. 5
  4. 1
  5. 8
  6. 3
  7. 2
  8. 1
  9. 4
  10. 6
  11. 1
  12. 2
  13. 2511
  14. 2
  15. 3
  16. 1
  17. 1
  18. 1
  19. 2934
  20. 2280
  21. 7
  22. 1
  23. 3
  24. 1
  25. 4
  26. 4
  27. 4
  28. 4
  29. 7
  30. 1575
  31. 1692
  32. 1377
  33. 4
  34. 2
  35. 2
  36. 4
  37. 3
  38. 9
  39. 9
  40. 8

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