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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 108 · 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 471 to make it divisible by 8? (a) 2 (b) 7 (c) 1 (d) 3
  2. 2.What is the smallest number you must add to 324 to make it divisible by 10? (a) 4 (b) 7 (c) 5 (d) 6
  3. 3.What is the units digit of 8^47? (a) 4 (b) 8 (c) 6 (d) 2
  4. 4.What is the smallest number you must add to 163 to make it divisible by 11? (a) 9 (b) 1 (c) 3 (d) 2
  5. 5.What is the smallest number you must add to 139 to make it divisible by 3? (a) 2 (b) 1
  6. 6.Which of these numbers is divisible by 8? (a) 6219 (b) 6214 (c) 6216 (d) 6226
  7. 7.What is the smallest number you must add to 101 to make it divisible by 3? (a) 2 (b) 1
  8. 8.What is the units digit of 3^43? (a) 9 (b) 1 (c) 3 (d) 7
  9. 9.What is the units digit of 8^33? (a) 4 (b) 2 (c) 8 (d) 6
  10. 10.What is the units digit of 2^28? (a) 4 (b) 8 (c) 6 (d) 2
  11. 11.What is the remainder when 374 is divided by 12? (a) 10 (b) 1 (c) 2 (d) 3
  12. 12.What is the smallest number you must add to 652 to make it divisible by 12? (a) 9 (b) 8 (c) 4 (d) 7
  13. 13.What is the remainder when 365 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 6
  14. 14.What is the units digit of 2^12? (a) 6 (b) 2 (c) 4 (d) 8
  15. 15.What is the units digit of 9^54? (a) 1 (b) 2 (c) 9 (d) 0
  16. 16.What is the units digit of 3^52? (a) 9 (b) 1 (c) 3 (d) 7
  17. 17.What is the smallest number you must add to 56 to make it divisible by 3? (a) 2 (b) 1
  18. 18.Which of these numbers is divisible by 4? (a) 2241 (b) 2233 (c) 2237 (d) 2236
  19. 19.What is the units digit of 2^40? (a) 2 (b) 6 (c) 4 (d) 8
  20. 20.What is the units digit of 2^51? (a) 2 (b) 4 (c) 6 (d) 8
  21. 21.What is the units digit of 2^45? (a) 6 (b) 2 (c) 4 (d) 8
  22. 22.What is the remainder when 347 is divided by 12? (a) 11 (b) 1 (c) 10 (d) 9
  23. 23.What is the units digit of 3^29? (a) 1 (b) 9 (c) 3 (d) 7
  24. 24.What is the units digit of 4^47? (a) 3 (b) 4 (c) 5 (d) 6
  25. 25.What is the smallest number you must add to 454 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
  26. 26.Which of these numbers is divisible by 4? (a) 1448 (b) 1447 (c) 1455 (d) 1451
  27. 27.Which of these numbers is divisible by 6? (a) 5373 (b) 5365 (c) 5364 (d) 5361
  28. 28.What is the remainder when 616 is divided by 10? (a) 5 (b) 4 (c) 7 (d) 6
  29. 29.Which of these numbers is divisible by 8? (a) 5384 (b) 5394 (c) 5377 (d) 5390
  30. 30.What is the smallest number you must add to 738 to make it divisible by 12? (a) 6 (b) 8 (c) 7 (d) 5
  31. 31.What is the remainder when 203 is divided by 3? (a) 1 (b) 2 (c) 0
  32. 32.What is the units digit of 4^54? (a) 6 (b) 7 (c) 5 (d) 4
  33. 33.What is the smallest number you must add to 604 to make it divisible by 9? (a) 7 (b) 8 (c) 6 (d) 1
  34. 34.What is the remainder when 351 is divided by 12? (a) 4 (b) 9 (c) 2 (d) 3
  35. 35.What is the remainder when 85 is divided by 3? (a) 1 (b) 0 (c) 2
  36. 36.Which of these numbers is divisible by 4? (a) 433 (b) 442 (c) 436 (d) 438
  37. 37.Which of these numbers is divisible by 4? (a) 3193 (b) 3201 (c) 3196 (d) 3197
  38. 38.Which of these numbers is divisible by 3? (a) 2323 (b) 2327 (c) 2322 (d) 2320
  39. 39.Which of these numbers is divisible by 8? (a) 7096 (b) 7091 (c) 7101 (d) 7105
  40. 40.What is the remainder when 139 is divided by 3? (a) 1 (b) 0 (c) 2

Answer key

  1. 1
  2. 6
  3. 2
  4. 2
  5. 2
  6. 6216
  7. 1
  8. 7
  9. 8
  10. 6
  11. 2
  12. 8
  13. 5
  14. 6
  15. 1
  16. 1
  17. 1
  18. 2236
  19. 6
  20. 8
  21. 2
  22. 11
  23. 3
  24. 4
  25. 2
  26. 1448
  27. 5364
  28. 6
  29. 5384
  30. 6
  31. 2
  32. 6
  33. 8
  34. 3
  35. 1
  36. 436
  37. 3196
  38. 2322
  39. 7096
  40. 1

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