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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 935 · 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 345 to make it divisible by 8? (a) 6 (b) 1 (c) 5 (d) 7
  2. 2.Which of these numbers is divisible by 3? (a) 330 (b) 328 (c) 334 (d) 332
  3. 3.What is the smallest number you must add to 278 to make it divisible by 10? (a) 2 (b) 3 (c) 8 (d) 1
  4. 4.What is the remainder when 286 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
  5. 5.What is the smallest number you must add to 181 to make it divisible by 11? (a) 8 (b) 5 (c) 7 (d) 6
  6. 6.Which of these numbers is divisible by 6? (a) 3846 (b) 3855 (c) 3851 (d) 3843
  7. 7.What is the remainder when 753 is divided by 11? (a) 6 (b) 4 (c) 5 (d) 7
  8. 8.Which of these numbers is divisible by 11? (a) 2243 (b) 2217 (c) 2222 (d) 2229
  9. 9.What is the smallest number you must add to 573 to make it divisible by 10? (a) 8 (b) 7 (c) 3 (d) 6
  10. 10.What is the units digit of 3^35? (a) 7 (b) 9 (c) 1 (d) 3
  11. 11.Which of these numbers is divisible by 6? (a) 3113 (b) 3107 (c) 3102 (d) 3097
  12. 12.Which of these numbers is divisible by 6? (a) 1187 (b) 1179 (c) 1173 (d) 1176
  13. 13.What is the smallest number you must add to 165 to make it divisible by 10? (a) 5 (b) 7 (c) 6 (d) 4
  14. 14.What is the remainder when 298 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 2
  15. 15.What is the smallest number you must add to 277 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  16. 16.What is the remainder when 574 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7
  17. 17.What is the smallest number you must add to 53 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
  18. 18.What is the units digit of 7^25? (a) 3 (b) 7 (c) 1 (d) 9
  19. 19.Which of these numbers is divisible by 9? (a) 6518 (b) 6538 (c) 6531 (d) 6525
  20. 20.What is the units digit of 7^53? (a) 1 (b) 3 (c) 9 (d) 7
  21. 21.What is the remainder when 102 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
  22. 22.Which of these numbers is divisible by 4? (a) 2496 (b) 2503 (c) 2494 (d) 2499
  23. 23.What is the units digit of 8^11? (a) 4 (b) 6 (c) 8 (d) 2
  24. 24.What is the units digit of 4^27? (a) 6 (b) 5 (c) 4 (d) 3
  25. 25.What is the units digit of 4^22? (a) 6 (b) 7 (c) 4 (d) 5
  26. 26.What is the units digit of 3^16? (a) 3 (b) 9 (c) 7 (d) 1
  27. 27.What is the units digit of 8^51? (a) 8 (b) 4 (c) 6 (d) 2
  28. 28.What is the smallest number you must add to 304 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
  29. 29.Which of these numbers is divisible by 4? (a) 510 (b) 508 (c) 514 (d) 505
  30. 30.What is the remainder when 429 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
  31. 31.What is the remainder when 385 is divided by 8? (a) 1 (b) 2 (c) 0 (d) 7
  32. 32.What is the smallest number you must add to 298 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
  33. 33.What is the smallest number you must add to 269 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  34. 34.What is the remainder when 413 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 5
  35. 35.What is the remainder when 279 is divided by 12? (a) 9 (b) 2 (c) 4 (d) 3
  36. 36.What is the remainder when 110 is divided by 7? (a) 2 (b) 6 (c) 4 (d) 5
  37. 37.What is the smallest number you must add to 166 to make it divisible by 3? (a) 2 (b) 1
  38. 38.What is the smallest number you must add to 328 to make it divisible by 11? (a) 3 (b) 2 (c) 1 (d) 9
  39. 39.What is the smallest number you must add to 162 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 1
  40. 40.Which of these numbers is divisible by 3? (a) 421 (b) 420 (c) 424 (d) 419

Answer key

  1. 7
  2. 330
  3. 2
  4. 1
  5. 6
  6. 3846
  7. 5
  8. 2222
  9. 7
  10. 7
  11. 3102
  12. 1176
  13. 5
  14. 4
  15. 3
  16. 6
  17. 2
  18. 7
  19. 6525
  20. 7
  21. 2
  22. 2496
  23. 2
  24. 4
  25. 6
  26. 1
  27. 2
  28. 1
  29. 508
  30. 2
  31. 1
  32. 6
  33. 3
  34. 5
  35. 3
  36. 5
  37. 2
  38. 2
  39. 6
  40. 420

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