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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 557 · 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 134 to make it divisible by 10? (a) 5 (b) 6 (c) 7 (d) 4
  2. 2.What is the units digit of 4^59? (a) 6 (b) 5 (c) 3 (d) 4
  3. 3.What is the smallest number you must add to 131 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  4. 4.What is the remainder when 81 is divided by 5? (a) 1 (b) 0 (c) 2 (d) 4
  5. 5.What is the remainder when 225 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
  6. 6.What is the smallest number you must add to 204 to make it divisible by 10? (a) 6 (b) 5 (c) 4 (d) 7
  7. 7.What is the units digit of 4^47? (a) 6 (b) 5 (c) 3 (d) 4
  8. 8.What is the units digit of 9^23? (a) 8 (b) 9 (c) 7 (d) 1
  9. 9.What is the smallest number you must add to 279 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
  10. 10.What is the remainder when 78 is divided by 7? (a) 1 (b) 0 (c) 6 (d) 2
  11. 11.What is the smallest number you must add to 356 to make it divisible by 12? (a) 8 (b) 3 (c) 4 (d) 5
  12. 12.What is the smallest number you must add to 394 to make it divisible by 12? (a) 10 (b) 1 (c) 3 (d) 2
  13. 13.Which of these numbers is divisible by 6? (a) 359 (b) 361 (c) 351 (d) 354
  14. 14.What is the smallest number you must add to 187 to make it divisible by 3? (a) 1 (b) 2
  15. 15.Which of these numbers is divisible by 6? (a) 498 (b) 503 (c) 494 (d) 507
  16. 16.What is the smallest number you must add to 809 to make it divisible by 12? (a) 5 (b) 8 (c) 7 (d) 6
  17. 17.What is the smallest number you must add to 199 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
  18. 18.Which of these numbers is divisible by 11? (a) 2549 (b) 2525 (c) 2535 (d) 2530
  19. 19.What is the remainder when 633 is divided by 10? (a) 4 (b) 2 (c) 3 (d) 7
  20. 20.What is the remainder when 609 is divided by 12? (a) 9 (b) 10 (c) 3 (d) 8
  21. 21.What is the smallest number you must add to 328 to make it divisible by 7? (a) 6 (b) 3 (c) 1 (d) 2
  22. 22.Which of these numbers is divisible by 3? (a) 512 (b) 510 (c) 514 (d) 508
  23. 23.What is the smallest number you must add to 105 to make it divisible by 10? (a) 7 (b) 6 (c) 4 (d) 5
  24. 24.What is the remainder when 411 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
  25. 25.What is the units digit of 8^51? (a) 4 (b) 2 (c) 6 (d) 8
  26. 26.What is the units digit of 2^16? (a) 2 (b) 6 (c) 8 (d) 4
  27. 27.What is the smallest number you must add to 298 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  28. 28.What is the smallest number you must add to 316 to make it divisible by 9? (a) 7 (b) 6 (c) 1 (d) 8
  29. 29.What is the remainder when 68 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
  30. 30.What is the smallest number you must add to 242 to make it divisible by 9? (a) 1 (b) 8 (c) 2 (d) 3
  31. 31.What is the units digit of 2^23? (a) 4 (b) 6 (c) 8 (d) 2
  32. 32.Which of these numbers is divisible by 3? (a) 1392 (b) 1396 (c) 1390 (d) 1394
  33. 33.What is the smallest number you must add to 395 to make it divisible by 12? (a) 1 (b) 2 (c) 3 (d) 11
  34. 34.What is the units digit of 9^14? (a) 2 (b) 9 (c) 1 (d) 0
  35. 35.What is the units digit of 9^32? (a) 0 (b) 9 (c) 1 (d) 2
  36. 36.Which of these numbers is divisible by 11? (a) 8975 (b) 8983 (c) 8965 (d) 8957
  37. 37.What is the units digit of 8^42? (a) 8 (b) 2 (c) 4 (d) 6
  38. 38.What is the remainder when 346 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
  39. 39.What is the smallest number you must add to 172 to make it divisible by 7? (a) 4 (b) 3 (c) 5 (d) 2
  40. 40.What is the smallest number you must add to 374 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 3

Answer key

  1. 6
  2. 4
  3. 1
  4. 1
  5. 1
  6. 6
  7. 4
  8. 9
  9. 3
  10. 1
  11. 4
  12. 2
  13. 354
  14. 2
  15. 498
  16. 7
  17. 5
  18. 2530
  19. 3
  20. 9
  21. 1
  22. 510
  23. 5
  24. 3
  25. 2
  26. 6
  27. 2
  28. 8
  29. 3
  30. 1
  31. 8
  32. 1392
  33. 1
  34. 1
  35. 1
  36. 8965
  37. 4
  38. 1
  39. 3
  40. 4

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