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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 38 · 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) 1062 (b) 1061 (c) 1066 (d) 1063
  2. 2.What is the units digit of 7^24? (a) 1 (b) 3 (c) 9 (d) 7
  3. 3.What is the smallest number you must add to 786 to make it divisible by 10? (a) 5 (b) 3 (c) 4 (d) 6
  4. 4.What is the remainder when 47 is divided by 3? (a) 0 (b) 1 (c) 2
  5. 5.What is the units digit of 4^20? (a) 5 (b) 7 (c) 4 (d) 6
  6. 6.What is the remainder when 144 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
  7. 7.What is the smallest number you must add to 205 to make it divisible by 10? (a) 7 (b) 6 (c) 5 (d) 4
  8. 8.What is the smallest number you must add to 295 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 5
  9. 9.What is the remainder when 96 is divided by 5? (a) 2 (b) 0 (c) 1 (d) 4
  10. 10.What is the units digit of 2^52? (a) 4 (b) 8 (c) 2 (d) 6
  11. 11.Which of these numbers is divisible by 6? (a) 2797 (b) 2804 (c) 2812 (d) 2802
  12. 12.Which of these numbers is divisible by 3? (a) 2092 (b) 2088 (c) 2089 (d) 2087
  13. 13.Which of these numbers is divisible by 8? (a) 4947 (b) 4958 (c) 4938 (d) 4944
  14. 14.What is the remainder when 421 is divided by 6? (a) 5 (b) 0 (c) 1 (d) 2
  15. 15.What is the smallest number you must add to 324 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 2
  16. 16.What is the smallest number you must add to 229 to make it divisible by 7? (a) 3 (b) 5 (c) 1 (d) 2
  17. 17.What is the units digit of 4^56? (a) 6 (b) 5 (c) 7 (d) 4
  18. 18.What is the smallest number you must add to 269 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 5
  19. 19.Which of these numbers is divisible by 3? (a) 1957 (b) 1953 (c) 1954 (d) 1952
  20. 20.What is the smallest number you must add to 812 to make it divisible by 12? (a) 5 (b) 3 (c) 8 (d) 4
  21. 21.Which of these numbers is divisible by 4? (a) 554 (b) 551 (c) 559 (d) 552
  22. 22.Which of these numbers is divisible by 9? (a) 3294 (b) 3300 (c) 3309 (d) 3289
  23. 23.What is the smallest number you must add to 37 to make it divisible by 3? (a) 1 (b) 2
  24. 24.What is the units digit of 4^34? (a) 5 (b) 6 (c) 4 (d) 7
  25. 25.What is the smallest number you must add to 238 to make it divisible by 3? (a) 2 (b) 1
  26. 26.What is the remainder when 138 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
  27. 27.What is the units digit of 8^34? (a) 2 (b) 6 (c) 4 (d) 8
  28. 28.What is the smallest number you must add to 471 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 2
  29. 29.Which of these numbers is divisible by 3? (a) 1106 (b) 1109 (c) 1112 (d) 1107
  30. 30.What is the units digit of 3^12? (a) 9 (b) 7 (c) 3 (d) 1
  31. 31.Which of these numbers is divisible by 4? (a) 1502 (b) 1493 (c) 1496 (d) 1498
  32. 32.What is the smallest number you must add to 93 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
  33. 33.Which of these numbers is divisible by 6? (a) 4053 (b) 4061 (c) 4056 (d) 4063
  34. 34.What is the units digit of 7^31? (a) 9 (b) 3 (c) 1 (d) 7
  35. 35.What is the smallest number you must add to 541 to make it divisible by 11? (a) 9 (b) 10 (c) 2 (d) 8
  36. 36.What is the smallest number you must add to 411 to make it divisible by 12? (a) 3 (b) 9 (c) 8 (d) 10
  37. 37.Which of these numbers is divisible by 3? (a) 366 (b) 365 (c) 371 (d) 368
  38. 38.What is the units digit of 3^44? (a) 1 (b) 3 (c) 7 (d) 9
  39. 39.What is the smallest number you must add to 416 to make it divisible by 10? (a) 6 (b) 5 (c) 4 (d) 3
  40. 40.What is the remainder when 410 is divided by 11? (a) 3 (b) 8 (c) 4 (d) 2

Answer key

  1. 1062
  2. 1
  3. 4
  4. 2
  5. 6
  6. 4
  7. 5
  8. 5
  9. 1
  10. 6
  11. 2802
  12. 2088
  13. 4944
  14. 1
  15. 5
  16. 2
  17. 6
  18. 1
  19. 1953
  20. 4
  21. 552
  22. 3294
  23. 2
  24. 6
  25. 2
  26. 2
  27. 4
  28. 3
  29. 1107
  30. 1
  31. 1496
  32. 3
  33. 4056
  34. 3
  35. 9
  36. 9
  37. 366
  38. 1
  39. 4
  40. 3

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