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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 461 · 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 294 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  2. 2.What is the smallest number you must add to 94 to make it divisible by 6? (a) 2 (b) 4 (c) 1 (d) 3
  3. 3.What is the smallest number you must add to 400 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 2
  4. 4.What is the smallest number you must add to 396 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
  5. 5.Which of these numbers is divisible by 8? (a) 2089 (b) 2110 (c) 2096 (d) 2101
  6. 6.Which of these numbers is divisible by 4? (a) 1366 (b) 1362 (c) 1360 (d) 1358
  7. 7.Which of these numbers is divisible by 9? (a) 3583 (b) 3594 (c) 3582 (d) 3581
  8. 8.Which of these numbers is divisible by 4? (a) 915 (b) 907 (c) 909 (d) 908
  9. 9.What is the units digit of 8^51? (a) 4 (b) 8 (c) 6 (d) 2
  10. 10.What is the units digit of 7^15? (a) 9 (b) 1 (c) 7 (d) 3
  11. 11.What is the remainder when 400 is divided by 7? (a) 2 (b) 0 (c) 1 (d) 6
  12. 12.What is the remainder when 173 is divided by 3? (a) 0 (b) 1 (c) 2
  13. 13.Which of these numbers is divisible by 3? (a) 2387 (b) 2392 (c) 2390 (d) 2388
  14. 14.What is the remainder when 129 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  15. 15.What is the remainder when 90 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
  16. 16.Which of these numbers is divisible by 9? (a) 7236 (b) 7246 (c) 7241 (d) 7234
  17. 17.What is the remainder when 246 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
  18. 18.Which of these numbers is divisible by 3? (a) 964 (b) 961 (c) 963 (d) 967
  19. 19.Which of these numbers is divisible by 6? (a) 4336 (b) 4325 (c) 4327 (d) 4326
  20. 20.What is the smallest number you must add to 284 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 6
  21. 21.What is the smallest number you must add to 94 to make it divisible by 7? (a) 4 (b) 5 (c) 3 (d) 6
  22. 22.Which of these numbers is divisible by 8? (a) 3208 (b) 3217 (c) 3205 (d) 3209
  23. 23.Which of these numbers is divisible by 3? (a) 2319 (b) 2321 (c) 2318 (d) 2324
  24. 24.What is the remainder when 389 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
  25. 25.What is the remainder when 799 is divided by 12? (a) 7 (b) 6 (c) 8 (d) 5
  26. 26.What is the remainder when 726 is divided by 10? (a) 5 (b) 7 (c) 6 (d) 4
  27. 27.Which of these numbers is divisible by 11? (a) 8456 (b) 8448 (c) 8462 (d) 8445
  28. 28.What is the remainder when 309 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
  29. 29.What is the units digit of 4^40? (a) 6 (b) 7 (c) 5 (d) 4
  30. 30.What is the remainder when 865 is divided by 11? (a) 8 (b) 4 (c) 7 (d) 6
  31. 31.What is the remainder when 533 is divided by 10? (a) 3 (b) 4 (c) 2 (d) 7
  32. 32.What is the units digit of 2^15? (a) 8 (b) 6 (c) 4 (d) 2
  33. 33.Which of these numbers is divisible by 3? (a) 623 (b) 629 (c) 624 (d) 625
  34. 34.What is the smallest number you must add to 650 to make it divisible by 9? (a) 7 (b) 2 (c) 8 (d) 6
  35. 35.What is the smallest number you must add to 625 to make it divisible by 8? (a) 7 (b) 6 (c) 1 (d) 5
  36. 36.Which of these numbers is divisible by 6? (a) 2698 (b) 2690 (c) 2687 (d) 2688
  37. 37.Which of these numbers is divisible by 6? (a) 2254 (b) 2242 (c) 2246 (d) 2244
  38. 38.What is the smallest number you must add to 754 to make it divisible by 12? (a) 1 (b) 10 (c) 2 (d) 3
  39. 39.Which of these numbers is divisible by 6? (a) 632 (b) 629 (c) 620 (d) 624
  40. 40.What is the smallest number you must add to 358 to make it divisible by 7? (a) 4 (b) 1 (c) 5 (d) 6

Answer key

  1. 2
  2. 2
  3. 2
  4. 4
  5. 2096
  6. 1360
  7. 3582
  8. 908
  9. 2
  10. 3
  11. 1
  12. 2
  13. 2388
  14. 1
  15. 2
  16. 7236
  17. 2
  18. 963
  19. 4326
  20. 4
  21. 4
  22. 3208
  23. 2319
  24. 4
  25. 7
  26. 6
  27. 8448
  28. 4
  29. 6
  30. 7
  31. 3
  32. 8
  33. 624
  34. 7
  35. 7
  36. 2688
  37. 2244
  38. 2
  39. 624
  40. 6

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