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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 867 · 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 remainder when 297 is divided by 6? (a) 3 (b) 2 (c) 5 (d) 4
  2. 2.Which of these numbers is divisible by 9? (a) 7344 (b) 7346 (c) 7337 (d) 7355
  3. 3.What is the remainder when 555 is divided by 7? (a) 3 (b) 1 (c) 2 (d) 5
  4. 4.Which of these numbers is divisible by 4? (a) 2367 (b) 2374 (c) 2369 (d) 2368
  5. 5.What is the remainder when 508 is divided by 10? (a) 9 (b) 8 (c) 7 (d) 2
  6. 6.What is the remainder when 197 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
  7. 7.Which of these numbers is divisible by 4? (a) 2400 (b) 2406 (c) 2398 (d) 2403
  8. 8.What is the remainder when 450 is divided by 8? (a) 1 (b) 2 (c) 6 (d) 3
  9. 9.Which of these numbers is divisible by 6? (a) 1619 (b) 1611 (c) 1608 (d) 1606
  10. 10.What is the smallest number you must add to 163 to make it divisible by 3? (a) 1 (b) 2
  11. 11.What is the smallest number you must add to 135 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  12. 12.What is the smallest number you must add to 188 to make it divisible by 9? (a) 1 (b) 2 (c) 3 (d) 8
  13. 13.What is the smallest number you must add to 87 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 3
  14. 14.What is the smallest number you must add to 276 to make it divisible by 9? (a) 2 (b) 6 (c) 4 (d) 3
  15. 15.What is the remainder when 310 is divided by 9? (a) 5 (b) 3 (c) 6 (d) 4
  16. 16.What is the smallest number you must add to 114 to make it divisible by 10? (a) 6 (b) 5 (c) 4 (d) 7
  17. 17.What is the remainder when 293 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
  18. 18.Which of these numbers is divisible by 6? (a) 1569 (b) 1580 (c) 1575 (d) 1572
  19. 19.What is the remainder when 34 is divided by 3? (a) 2 (b) 0 (c) 1
  20. 20.Which of these numbers is divisible by 11? (a) 947 (b) 936 (c) 925 (d) 935
  21. 21.Which of these numbers is divisible by 11? (a) 1519 (b) 1518 (c) 1508 (d) 1530
  22. 22.What is the remainder when 217 is divided by 9? (a) 8 (b) 2 (c) 1 (d) 0
  23. 23.What is the remainder when 178 is divided by 8? (a) 6 (b) 2 (c) 3 (d) 1
  24. 24.Which of these numbers is divisible by 9? (a) 1452 (b) 1436 (c) 1448 (d) 1440
  25. 25.What is the remainder when 306 is divided by 8? (a) 3 (b) 2 (c) 6 (d) 1
  26. 26.What is the remainder when 126 is divided by 5? (a) 1 (b) 0 (c) 4 (d) 2
  27. 27.What is the remainder when 152 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 1
  28. 28.Which of these numbers is divisible by 3? (a) 2294 (b) 2292 (c) 2297 (d) 2290
  29. 29.What is the smallest number you must add to 121 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 4
  30. 30.Which of these numbers is divisible by 11? (a) 6765 (b) 6764 (c) 6786 (d) 6770
  31. 31.What is the units digit of 3^19? (a) 9 (b) 1 (c) 7 (d) 3
  32. 32.Which of these numbers is divisible by 6? (a) 260 (b) 269 (c) 264 (d) 272
  33. 33.What is the smallest number you must add to 367 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
  34. 34.What is the smallest number you must add to 149 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 1
  35. 35.Which of these numbers is divisible by 4? (a) 2982 (b) 2990 (c) 2984 (d) 2987
  36. 36.What is the smallest number you must add to 112 to make it divisible by 6? (a) 4 (b) 1 (c) 2 (d) 3
  37. 37.What is the smallest number you must add to 262 to make it divisible by 10? (a) 7 (b) 8 (c) 2 (d) 9
  38. 38.What is the smallest number you must add to 572 to make it divisible by 10? (a) 9 (b) 2 (c) 7 (d) 8
  39. 39.Which of these numbers is divisible by 3? (a) 380 (b) 381 (c) 385 (d) 383
  40. 40.Which of these numbers is divisible by 9? (a) 1215 (b) 1221 (c) 1229 (d) 1212

Answer key

  1. 3
  2. 7344
  3. 2
  4. 2368
  5. 8
  6. 2
  7. 2400
  8. 2
  9. 1608
  10. 2
  11. 1
  12. 1
  13. 4
  14. 3
  15. 4
  16. 6
  17. 1
  18. 1572
  19. 1
  20. 935
  21. 1518
  22. 1
  23. 2
  24. 1440
  25. 2
  26. 1
  27. 2
  28. 2292
  29. 5
  30. 6765
  31. 7
  32. 264
  33. 3
  34. 1
  35. 2984
  36. 2
  37. 8
  38. 8
  39. 381
  40. 1215

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