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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 538 · 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 3? (a) 644 (b) 645 (c) 647 (d) 650
  2. 2.What is the smallest number you must add to 123 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
  3. 3.What is the remainder when 487 is divided by 8? (a) 6 (b) 1 (c) 7 (d) 5
  4. 4.Which of these numbers is divisible by 9? (a) 1344 (b) 1356 (c) 1341 (d) 1337
  5. 5.Which of these numbers is divisible by 3? (a) 1449 (b) 1454 (c) 1451 (d) 1448
  6. 6.What is the smallest number you must add to 233 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  7. 7.What is the remainder when 165 is divided by 9? (a) 3 (b) 4 (c) 6 (d) 2
  8. 8.Which of these numbers is divisible by 3? (a) 2557 (b) 2559 (c) 2564 (d) 2561
  9. 9.What is the smallest number you must add to 239 to make it divisible by 10? (a) 3 (b) 9 (c) 2 (d) 1
  10. 10.Which of these numbers is divisible by 6? (a) 3216 (b) 3211 (c) 3227 (d) 3217
  11. 11.What is the remainder when 785 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 7
  12. 12.What is the units digit of 8^24? (a) 2 (b) 8 (c) 6 (d) 4
  13. 13.What is the smallest number you must add to 347 to make it divisible by 8? (a) 6 (b) 5 (c) 3 (d) 4
  14. 14.Which of these numbers is divisible by 4? (a) 2451 (b) 2447 (c) 2448 (d) 2455
  15. 15.Which of these numbers is divisible by 11? (a) 5844 (b) 5853 (c) 5831 (d) 5841
  16. 16.What is the smallest number you must add to 615 to make it divisible by 11? (a) 10 (b) 2 (c) 3 (d) 1
  17. 17.Which of these numbers is divisible by 3? (a) 1662 (b) 1660 (c) 1663 (d) 1667
  18. 18.What is the remainder when 52 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
  19. 19.What is the remainder when 410 is divided by 9? (a) 4 (b) 6 (c) 7 (d) 5
  20. 20.Which of these numbers is divisible by 4? (a) 1882 (b) 1884 (c) 1885 (d) 1889
  21. 21.What is the smallest number you must add to 626 to make it divisible by 9? (a) 5 (b) 6 (c) 3 (d) 4
  22. 22.Which of these numbers is divisible by 4? (a) 3359 (b) 3366 (c) 3361 (d) 3360
  23. 23.What is the remainder when 649 is divided by 12? (a) 0 (b) 1 (c) 2 (d) 11
  24. 24.What is the remainder when 145 is divided by 8? (a) 0 (b) 7 (c) 1 (d) 2
  25. 25.Which of these numbers is divisible by 3? (a) 193 (b) 190 (c) 187 (d) 189
  26. 26.What is the smallest number you must add to 318 to make it divisible by 12? (a) 7 (b) 5 (c) 6 (d) 8
  27. 27.What is the remainder when 494 is divided by 10? (a) 5 (b) 6 (c) 4 (d) 3
  28. 28.Which of these numbers is divisible by 9? (a) 4302 (b) 4305 (c) 4298 (d) 4312
  29. 29.What is the smallest number you must add to 483 to make it divisible by 9? (a) 6 (b) 4 (c) 2 (d) 3
  30. 30.What is the smallest number you must add to 116 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
  31. 31.Which of these numbers is divisible by 8? (a) 780 (b) 768 (c) 767 (d) 769
  32. 32.What is the smallest number you must add to 339 to make it divisible by 10? (a) 1 (b) 2 (c) 9 (d) 3
  33. 33.Which of these numbers is divisible by 4? (a) 1256 (b) 1261 (c) 1254 (d) 1258
  34. 34.What is the units digit of 2^46? (a) 2 (b) 6 (c) 4 (d) 8
  35. 35.What is the remainder when 230 is divided by 11? (a) 10 (b) 8 (c) 1 (d) 9
  36. 36.What is the units digit of 7^40? (a) 7 (b) 9 (c) 1 (d) 3
  37. 37.What is the units digit of 3^41? (a) 3 (b) 9 (c) 7 (d) 1
  38. 38.Which of these numbers is divisible by 3? (a) 290 (b) 295 (c) 293 (d) 291
  39. 39.What is the remainder when 478 is divided by 7? (a) 5 (b) 2 (c) 1 (d) 3
  40. 40.What is the smallest number you must add to 133 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 5

Answer key

  1. 645
  2. 2
  3. 7
  4. 1341
  5. 1449
  6. 3
  7. 3
  8. 2559
  9. 1
  10. 3216
  11. 5
  12. 6
  13. 5
  14. 2448
  15. 5841
  16. 1
  17. 1662
  18. 2
  19. 5
  20. 1884
  21. 4
  22. 3360
  23. 1
  24. 1
  25. 189
  26. 6
  27. 4
  28. 4302
  29. 3
  30. 4
  31. 768
  32. 1
  33. 1256
  34. 4
  35. 10
  36. 1
  37. 3
  38. 291
  39. 2
  40. 5

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