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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 492 · 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 11? (a) 5191 (b) 5175 (c) 5181 (d) 5199
  2. 2.What is the smallest number you must add to 496 to make it divisible by 7? (a) 2 (b) 1 (c) 6 (d) 3
  3. 3.Which of these numbers is divisible by 8? (a) 932 (b) 941 (c) 936 (d) 951
  4. 4.Which of these numbers is divisible by 6? (a) 4457 (b) 4469 (c) 4461 (d) 4458
  5. 5.What is the units digit of 9^50? (a) 0 (b) 2 (c) 1 (d) 9
  6. 6.What is the remainder when 142 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
  7. 7.What is the smallest number you must add to 366 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
  8. 8.What is the units digit of 3^10? (a) 7 (b) 1 (c) 9 (d) 3
  9. 9.What is the remainder when 724 is divided by 10? (a) 5 (b) 3 (c) 6 (d) 4
  10. 10.What is the smallest number you must add to 61 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
  11. 11.What is the smallest number you must add to 315 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  12. 12.What is the remainder when 63 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
  13. 13.What is the units digit of 8^54? (a) 8 (b) 4 (c) 2 (d) 6
  14. 14.What is the remainder when 95 is divided by 9? (a) 4 (b) 5 (c) 6 (d) 7
  15. 15.Which of these numbers is divisible by 4? (a) 1023 (b) 1020 (c) 1027 (d) 1019
  16. 16.Which of these numbers is divisible by 4? (a) 3051 (b) 3045 (c) 3044 (d) 3043
  17. 17.What is the smallest number you must add to 73 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  18. 18.What is the smallest number you must add to 119 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  19. 19.What is the smallest number you must add to 943 to make it divisible by 12? (a) 7 (b) 5 (c) 6 (d) 4
  20. 20.What is the smallest number you must add to 456 to make it divisible by 7? (a) 4 (b) 6 (c) 1 (d) 5
  21. 21.What is the smallest number you must add to 604 to make it divisible by 12? (a) 9 (b) 8 (c) 7 (d) 4
  22. 22.What is the remainder when 413 is divided by 9? (a) 1 (b) 8 (c) 7 (d) 6
  23. 23.Which of these numbers is divisible by 6? (a) 268 (b) 258 (c) 263 (d) 257
  24. 24.Which of these numbers is divisible by 3? (a) 683 (b) 680 (c) 676 (d) 678
  25. 25.What is the smallest number you must add to 207 to make it divisible by 8? (a) 7 (b) 3 (c) 2 (d) 1
  26. 26.Which of these numbers is divisible by 8? (a) 3768 (b) 3765 (c) 3778 (d) 3772
  27. 27.What is the remainder when 209 is divided by 12? (a) 4 (b) 5 (c) 7 (d) 6
  28. 28.What is the smallest number you must add to 118 to make it divisible by 3? (a) 1 (b) 2
  29. 29.Which of these numbers is divisible by 11? (a) 1504 (b) 1510 (c) 1526 (d) 1507
  30. 30.Which of these numbers is divisible by 6? (a) 3036 (b) 3041 (c) 3046 (d) 3034
  31. 31.What is the units digit of 2^14? (a) 6 (b) 2 (c) 4 (d) 8
  32. 32.What is the remainder when 130 is divided by 3? (a) 0 (b) 2 (c) 1
  33. 33.What is the remainder when 121 is divided by 12? (a) 11 (b) 2 (c) 0 (d) 1
  34. 34.Which of these numbers is divisible by 8? (a) 4069 (b) 4058 (c) 4064 (d) 4078
  35. 35.What is the units digit of 9^39? (a) 7 (b) 1 (c) 9 (d) 8
  36. 36.What is the smallest number you must add to 208 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 2
  37. 37.What is the units digit of 8^50? (a) 6 (b) 8 (c) 2 (d) 4
  38. 38.What is the units digit of 9^52? (a) 0 (b) 9 (c) 1 (d) 2
  39. 39.What is the smallest number you must add to 288 to make it divisible by 7? (a) 6 (b) 1 (c) 4 (d) 5
  40. 40.What is the smallest number you must add to 74 to make it divisible by 4? (a) 1 (b) 2 (c) 3

Answer key

  1. 5181
  2. 1
  3. 936
  4. 4458
  5. 1
  6. 4
  7. 4
  8. 9
  9. 4
  10. 4
  11. 1
  12. 3
  13. 4
  14. 5
  15. 1020
  16. 3044
  17. 3
  18. 1
  19. 5
  20. 6
  21. 8
  22. 8
  23. 258
  24. 678
  25. 1
  26. 3768
  27. 5
  28. 2
  29. 1507
  30. 3036
  31. 4
  32. 1
  33. 1
  34. 4064
  35. 9
  36. 2
  37. 4
  38. 1
  39. 6
  40. 2

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