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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 665 · 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 287 to make it divisible by 8? (a) 2 (b) 7 (c) 1 (d) 3
  2. 2.Which of these numbers is divisible by 3? (a) 776 (b) 779 (c) 777 (d) 781
  3. 3.Which of these numbers is divisible by 3? (a) 2099 (b) 2096 (c) 2097 (d) 2101
  4. 4.What is the remainder when 152 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
  5. 5.What is the remainder when 198 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
  6. 6.What is the smallest number you must add to 199 to make it divisible by 12? (a) 6 (b) 4 (c) 7 (d) 5
  7. 7.What is the remainder when 386 is divided by 9? (a) 1 (b) 7 (c) 8 (d) 6
  8. 8.What is the smallest number you must add to 602 to make it divisible by 10? (a) 8 (b) 7 (c) 9 (d) 2
  9. 9.What is the units digit of 3^41? (a) 9 (b) 7 (c) 1 (d) 3
  10. 10.What is the smallest number you must add to 567 to make it divisible by 8? (a) 7 (b) 3 (c) 1 (d) 2
  11. 11.What is the remainder when 189 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
  12. 12.What is the smallest number you must add to 354 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
  13. 13.What is the smallest number you must add to 87 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  14. 14.What is the units digit of 3^32? (a) 7 (b) 9 (c) 3 (d) 1
  15. 15.Which of these numbers is divisible by 8? (a) 6578 (b) 6573 (c) 6563 (d) 6568
  16. 16.Which of these numbers is divisible by 6? (a) 2940 (b) 2947 (c) 2942 (d) 2938
  17. 17.What is the remainder when 178 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
  18. 18.Which of these numbers is divisible by 3? (a) 1320 (b) 1324 (c) 1321 (d) 1319
  19. 19.What is the smallest number you must add to 265 to make it divisible by 9? (a) 4 (b) 6 (c) 7 (d) 5
  20. 20.What is the units digit of 4^54? (a) 7 (b) 4 (c) 5 (d) 6
  21. 21.What is the smallest number you must add to 129 to make it divisible by 9? (a) 3 (b) 7 (c) 6 (d) 5
  22. 22.What is the remainder when 84 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
  23. 23.What is the remainder when 169 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
  24. 24.What is the remainder when 764 is divided by 10? (a) 6 (b) 5 (c) 3 (d) 4
  25. 25.What is the units digit of 3^53? (a) 1 (b) 7 (c) 3 (d) 9
  26. 26.What is the smallest number you must add to 298 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 4
  27. 27.What is the smallest number you must add to 122 to make it divisible by 3? (a) 2 (b) 1
  28. 28.Which of these numbers is divisible by 11? (a) 2242 (b) 2253 (c) 2256 (d) 2244
  29. 29.Which of these numbers is divisible by 4? (a) 3426 (b) 3419 (c) 3420 (d) 3421
  30. 30.What is the smallest number you must add to 263 to make it divisible by 11? (a) 1 (b) 2 (c) 10 (d) 3
  31. 31.Which of these numbers is divisible by 8? (a) 2228 (b) 2232 (c) 2241 (d) 2237
  32. 32.What is the remainder when 234 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
  33. 33.What is the smallest number you must add to 73 to make it divisible by 3? (a) 2 (b) 1
  34. 34.Which of these numbers is divisible by 9? (a) 4197 (b) 4211 (c) 4189 (d) 4194
  35. 35.What is the smallest number you must add to 105 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  36. 36.What is the smallest number you must add to 395 to make it divisible by 12? (a) 11 (b) 3 (c) 2 (d) 1
  37. 37.Which of these numbers is divisible by 8? (a) 383 (b) 364 (c) 373 (d) 368
  38. 38.What is the remainder when 668 is divided by 12? (a) 8 (b) 7 (c) 4 (d) 9
  39. 39.Which of these numbers is divisible by 8? (a) 3041 (b) 3055 (c) 3048 (d) 3057
  40. 40.Which of these numbers is divisible by 11? (a) 3373 (b) 3364 (c) 3366 (d) 3383

Answer key

  1. 1
  2. 777
  3. 2097
  4. 2
  5. 3
  6. 5
  7. 8
  8. 8
  9. 3
  10. 1
  11. 4
  12. 1
  13. 1
  14. 1
  15. 6568
  16. 2940
  17. 2
  18. 1320
  19. 5
  20. 6
  21. 6
  22. 4
  23. 1
  24. 4
  25. 3
  26. 2
  27. 1
  28. 2244
  29. 3420
  30. 1
  31. 2232
  32. 2
  33. 2
  34. 4194
  35. 3
  36. 1
  37. 368
  38. 8
  39. 3048
  40. 3366

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