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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 662 · 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 173 is divided by 9? (a) 3 (b) 7 (c) 1 (d) 2
  2. 2.Which of these numbers is divisible by 3? (a) 1334 (b) 1340 (c) 1337 (d) 1335
  3. 3.What is the remainder when 271 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
  4. 4.Which of these numbers is divisible by 6? (a) 413 (b) 402 (c) 399 (d) 407
  5. 5.Which of these numbers is divisible by 4? (a) 1439 (b) 1436 (c) 1441 (d) 1435
  6. 6.What is the units digit of 9^37? (a) 7 (b) 9 (c) 1 (d) 8
  7. 7.Which of these numbers is divisible by 3? (a) 736 (b) 743 (c) 739 (d) 738
  8. 8.What is the smallest number you must add to 494 to make it divisible by 7? (a) 4 (b) 2 (c) 3 (d) 5
  9. 9.Which of these numbers is divisible by 4? (a) 1672 (b) 1671 (c) 1673 (d) 1677
  10. 10.Which of these numbers is divisible by 4? (a) 1904 (b) 1911 (c) 1906 (d) 1902
  11. 11.What is the units digit of 7^23? (a) 1 (b) 9 (c) 3 (d) 7
  12. 12.What is the remainder when 492 is divided by 11? (a) 9 (b) 7 (c) 8 (d) 3
  13. 13.What is the smallest number you must add to 413 to make it divisible by 12? (a) 6 (b) 8 (c) 7 (d) 5
  14. 14.What is the smallest number you must add to 278 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  15. 15.What is the remainder when 178 is divided by 11? (a) 3 (b) 1 (c) 2 (d) 9
  16. 16.What is the smallest number you must add to 212 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
  17. 17.Which of these numbers is divisible by 6? (a) 1248 (b) 1258 (c) 1251 (d) 1244
  18. 18.Which of these numbers is divisible by 11? (a) 1536 (b) 1552 (c) 1540 (d) 1547
  19. 19.Which of these numbers is divisible by 3? (a) 1510 (b) 1508 (c) 1505 (d) 1506
  20. 20.Which of these numbers is divisible by 3? (a) 392 (b) 394 (c) 388 (d) 390
  21. 21.What is the remainder when 274 is divided by 12? (a) 2 (b) 11 (c) 9 (d) 10
  22. 22.Which of these numbers is divisible by 8? (a) 469 (b) 464 (c) 477 (d) 461
  23. 23.What is the units digit of 8^45? (a) 2 (b) 6 (c) 4 (d) 8
  24. 24.Which of these numbers is divisible by 8? (a) 1422 (b) 1413 (c) 1401 (d) 1408
  25. 25.What is the remainder when 262 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
  26. 26.What is the remainder when 317 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
  27. 27.What is the smallest number you must add to 521 to make it divisible by 7? (a) 6 (b) 3 (c) 5 (d) 4
  28. 28.What is the remainder when 139 is divided by 7? (a) 5 (b) 6 (c) 1 (d) 4
  29. 29.What is the smallest number you must add to 218 to make it divisible by 9? (a) 2 (b) 6 (c) 7 (d) 8
  30. 30.What is the smallest number you must add to 420 to make it divisible by 9? (a) 2 (b) 3 (c) 4 (d) 6
  31. 31.What is the remainder when 773 is divided by 12? (a) 6 (b) 4 (c) 7 (d) 5
  32. 32.Which of these numbers is divisible by 4? (a) 1712 (b) 1718 (c) 1715 (d) 1711
  33. 33.What is the remainder when 565 is divided by 9? (a) 6 (b) 8 (c) 2 (d) 7
  34. 34.What is the remainder when 577 is divided by 8? (a) 0 (b) 7 (c) 1 (d) 2
  35. 35.What is the units digit of 4^18? (a) 7 (b) 5 (c) 4 (d) 6
  36. 36.What is the smallest number you must add to 573 to make it divisible by 8? (a) 3 (b) 5 (c) 2 (d) 4
  37. 37.What is the remainder when 118 is divided by 11? (a) 8 (b) 3 (c) 9 (d) 7
  38. 38.What is the smallest number you must add to 669 to make it divisible by 10? (a) 9 (b) 3 (c) 1 (d) 2
  39. 39.What is the smallest number you must add to 104 to make it divisible by 9? (a) 3 (b) 6 (c) 5 (d) 4
  40. 40.Which of these numbers is divisible by 6? (a) 539 (b) 529 (c) 534 (d) 541

Answer key

  1. 2
  2. 1335
  3. 1
  4. 402
  5. 1436
  6. 9
  7. 738
  8. 3
  9. 1672
  10. 1904
  11. 3
  12. 8
  13. 7
  14. 2
  15. 2
  16. 4
  17. 1248
  18. 1540
  19. 1506
  20. 390
  21. 10
  22. 464
  23. 8
  24. 1408
  25. 2
  26. 1
  27. 4
  28. 6
  29. 7
  30. 3
  31. 5
  32. 1712
  33. 7
  34. 1
  35. 6
  36. 3
  37. 8
  38. 1
  39. 4
  40. 534

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