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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 154 · 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 298 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  2. 2.Which of these numbers is divisible by 8? (a) 2534 (b) 2542 (c) 2523 (d) 2528
  3. 3.Which of these numbers is divisible by 3? (a) 1885 (b) 1887 (c) 1891 (d) 1889
  4. 4.What is the units digit of 9^20? (a) 9 (b) 0 (c) 2 (d) 1
  5. 5.What is the units digit of 7^56? (a) 3 (b) 1 (c) 7 (d) 9
  6. 6.What is the remainder when 202 is divided by 9? (a) 6 (b) 3 (c) 5 (d) 4
  7. 7.What is the remainder when 328 is divided by 10? (a) 8 (b) 9 (c) 2 (d) 7
  8. 8.Which of these numbers is divisible by 3? (a) 1114 (b) 1113 (c) 1118 (d) 1111
  9. 9.What is the smallest number you must add to 225 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  10. 10.Which of these numbers is divisible by 6? (a) 2750 (b) 2755 (c) 2747 (d) 2748
  11. 11.What is the units digit of 2^12? (a) 2 (b) 6 (c) 4 (d) 8
  12. 12.What is the remainder when 957 is divided by 12? (a) 3 (b) 10 (c) 8 (d) 9
  13. 13.Which of these numbers is divisible by 11? (a) 807 (b) 815 (c) 814 (d) 827
  14. 14.Which of these numbers is divisible by 11? (a) 9393 (b) 9375 (c) 9372 (d) 9365
  15. 15.What is the remainder when 419 is divided by 11? (a) 10 (b) 0 (c) 2 (d) 1
  16. 16.What is the smallest number you must add to 121 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
  17. 17.Which of these numbers is divisible by 11? (a) 9079 (b) 9093 (c) 9075 (d) 9072
  18. 18.What is the smallest number you must add to 186 to make it divisible by 11? (a) 1 (b) 3 (c) 2 (d) 10
  19. 19.What is the smallest number you must add to 659 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 9
  20. 20.Which of these numbers is divisible by 4? (a) 2799 (b) 2793 (c) 2790 (d) 2792
  21. 21.Which of these numbers is divisible by 6? (a) 5263 (b) 5273 (c) 5261 (d) 5262
  22. 22.Which of these numbers is divisible by 6? (a) 4223 (b) 4232 (c) 4224 (d) 4227
  23. 23.What is the smallest number you must add to 244 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
  24. 24.What is the smallest number you must add to 157 to make it divisible by 3? (a) 2 (b) 1
  25. 25.What is the units digit of 4^46? (a) 7 (b) 5 (c) 4 (d) 6
  26. 26.Which of these numbers is divisible by 11? (a) 5911 (b) 5898 (c) 5907 (d) 5926
  27. 27.What is the remainder when 185 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 5
  28. 28.Which of these numbers is divisible by 9? (a) 1119 (b) 1108 (c) 1103 (d) 1107
  29. 29.What is the smallest number you must add to 50 to make it divisible by 3? (a) 1 (b) 2
  30. 30.Which of these numbers is divisible by 4? (a) 1404 (b) 1409 (c) 1405 (d) 1402
  31. 31.What is the smallest number you must add to 188 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
  32. 32.What is the remainder when 112 is divided by 3? (a) 0 (b) 2 (c) 1
  33. 33.Which of these numbers is divisible by 4? (a) 1635 (b) 1631 (c) 1628 (d) 1627
  34. 34.What is the remainder when 172 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
  35. 35.What is the remainder when 217 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
  36. 36.What is the remainder when 140 is divided by 6? (a) 3 (b) 1 (c) 4 (d) 2
  37. 37.What is the smallest number you must add to 263 to make it divisible by 12? (a) 1 (b) 3 (c) 2 (d) 11
  38. 38.Which of these numbers is divisible by 8? (a) 1135 (b) 1149 (c) 1142 (d) 1136
  39. 39.What is the remainder when 226 is divided by 10? (a) 5 (b) 6 (c) 7 (d) 4
  40. 40.What is the smallest number you must add to 316 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2

Answer key

  1. 2
  2. 2528
  3. 1887
  4. 1
  5. 1
  6. 4
  7. 8
  8. 1113
  9. 3
  10. 2748
  11. 6
  12. 9
  13. 814
  14. 9372
  15. 1
  16. 5
  17. 9075
  18. 1
  19. 1
  20. 2792
  21. 5262
  22. 4224
  23. 1
  24. 2
  25. 6
  26. 5907
  27. 5
  28. 1107
  29. 1
  30. 1404
  31. 4
  32. 1
  33. 1628
  34. 4
  35. 1
  36. 2
  37. 1
  38. 1136
  39. 6
  40. 4

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