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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 479 · 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 681 is divided by 9? (a) 3 (b) 7 (c) 6 (d) 5
  2. 2.Which of these numbers is divisible by 6? (a) 282 (b) 290 (c) 287 (d) 279
  3. 3.What is the remainder when 319 is divided by 6? (a) 0 (b) 5 (c) 1 (d) 2
  4. 4.What is the smallest number you must add to 388 to make it divisible by 12? (a) 9 (b) 4 (c) 7 (d) 8
  5. 5.Which of these numbers is divisible by 11? (a) 8030 (b) 8029 (c) 8032 (d) 8048
  6. 6.Which of these numbers is divisible by 6? (a) 3569 (b) 3562 (c) 3564 (d) 3575
  7. 7.What is the units digit of 3^34? (a) 3 (b) 9 (c) 7 (d) 1
  8. 8.What is the remainder when 97 is divided by 3? (a) 2 (b) 0 (c) 1
  9. 9.What is the smallest number you must add to 678 to make it divisible by 9? (a) 6 (b) 5 (c) 7 (d) 3
  10. 10.What is the smallest number you must add to 205 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  11. 11.What is the smallest number you must add to 178 to make it divisible by 10? (a) 8 (b) 2 (c) 1 (d) 3
  12. 12.Which of these numbers is divisible by 3? (a) 250 (b) 256 (c) 252 (d) 254
  13. 13.What is the units digit of 8^39? (a) 6 (b) 4 (c) 2 (d) 8
  14. 14.What is the smallest number you must add to 35 to make it divisible by 3? (a) 1 (b) 2
  15. 15.What is the smallest number you must add to 178 to make it divisible by 7? (a) 4 (b) 5 (c) 3 (d) 6
  16. 16.What is the remainder when 295 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
  17. 17.What is the remainder when 212 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
  18. 18.Which of these numbers is divisible by 6? (a) 3552 (b) 3553 (c) 3550 (d) 3563
  19. 19.What is the smallest number you must add to 324 to make it divisible by 11? (a) 6 (b) 8 (c) 5 (d) 7
  20. 20.Which of these numbers is divisible by 3? (a) 2246 (b) 2247 (c) 2249 (d) 2251
  21. 21.Which of these numbers is divisible by 3? (a) 1241 (b) 1236 (c) 1238 (d) 1234
  22. 22.What is the units digit of 7^60? (a) 1 (b) 9 (c) 7 (d) 3
  23. 23.What is the units digit of 3^40? (a) 1 (b) 9 (c) 7 (d) 3
  24. 24.What is the smallest number you must add to 249 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  25. 25.What is the smallest number you must add to 625 to make it divisible by 11? (a) 1 (b) 9 (c) 3 (d) 2
  26. 26.What is the remainder when 235 is divided by 9? (a) 2 (b) 8 (c) 1 (d) 0
  27. 27.What is the smallest number you must add to 861 to make it divisible by 11? (a) 3 (b) 8 (c) 9 (d) 7
  28. 28.Which of these numbers is divisible by 8? (a) 1636 (b) 1640 (c) 1650 (d) 1646
  29. 29.What is the units digit of 9^22? (a) 1 (b) 0 (c) 2 (d) 9
  30. 30.Which of these numbers is divisible by 6? (a) 2289 (b) 2297 (c) 2300 (d) 2292
  31. 31.What is the remainder when 85 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
  32. 32.What is the smallest number you must add to 596 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 6
  33. 33.What is the smallest number you must add to 315 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  34. 34.What is the smallest number you must add to 221 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
  35. 35.Which of these numbers is divisible by 4? (a) 2457 (b) 2454 (c) 2462 (d) 2456
  36. 36.Which of these numbers is divisible by 4? (a) 990 (b) 999 (c) 992 (d) 993
  37. 37.What is the units digit of 4^25? (a) 4 (b) 5 (c) 6 (d) 3
  38. 38.What is the units digit of 3^29? (a) 9 (b) 1 (c) 7 (d) 3
  39. 39.What is the smallest number you must add to 283 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  40. 40.Which of these numbers is divisible by 3? (a) 1948 (b) 1950 (c) 1952 (d) 1954

Answer key

  1. 6
  2. 282
  3. 1
  4. 8
  5. 8030
  6. 3564
  7. 9
  8. 1
  9. 6
  10. 3
  11. 2
  12. 252
  13. 2
  14. 1
  15. 4
  16. 1
  17. 2
  18. 3552
  19. 6
  20. 2247
  21. 1236
  22. 1
  23. 1
  24. 1
  25. 2
  26. 1
  27. 8
  28. 1640
  29. 1
  30. 2292
  31. 5
  32. 4
  33. 1
  34. 4
  35. 2456
  36. 992
  37. 4
  38. 3
  39. 1
  40. 1950

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