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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 716 · 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 11? (a) 9267 (b) 9282 (c) 9262 (d) 9260
  2. 2.Which of these numbers is divisible by 3? (a) 797 (b) 795 (c) 799 (d) 794
  3. 3.What is the remainder when 232 is divided by 10? (a) 2 (b) 3 (c) 8 (d) 1
  4. 4.What is the units digit of 4^18? (a) 6 (b) 5 (c) 4 (d) 7
  5. 5.What is the remainder when 247 is divided by 11? (a) 7 (b) 4 (c) 5 (d) 6
  6. 6.What is the smallest number you must add to 219 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  7. 7.What is the remainder when 925 is divided by 12? (a) 2 (b) 11 (c) 0 (d) 1
  8. 8.What is the remainder when 257 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
  9. 9.Which of these numbers is divisible by 6? (a) 2947 (b) 2959 (c) 2952 (d) 2954
  10. 10.What is the smallest number you must add to 367 to make it divisible by 10? (a) 3 (b) 2 (c) 4 (d) 7
  11. 11.What is the remainder when 555 is divided by 10? (a) 6 (b) 5 (c) 4 (d) 7
  12. 12.What is the smallest number you must add to 327 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
  13. 13.What is the smallest number you must add to 687 to make it divisible by 11? (a) 5 (b) 7 (c) 6 (d) 8
  14. 14.What is the smallest number you must add to 63 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  15. 15.What is the smallest number you must add to 362 to make it divisible by 10? (a) 2 (b) 8 (c) 7 (d) 9
  16. 16.Which of these numbers is divisible by 8? (a) 1990 (b) 1996 (c) 1982 (d) 1984
  17. 17.What is the remainder when 499 is divided by 12? (a) 8 (b) 6 (c) 7 (d) 5
  18. 18.What is the remainder when 395 is divided by 10? (a) 5 (b) 6 (c) 7 (d) 4
  19. 19.Which of these numbers is divisible by 3? (a) 1865 (b) 1859 (c) 1861 (d) 1860
  20. 20.What is the smallest number you must add to 203 to make it divisible by 3? (a) 2 (b) 1
  21. 21.What is the units digit of 4^57? (a) 4 (b) 5 (c) 6 (d) 3
  22. 22.Which of these numbers is divisible by 6? (a) 2338 (b) 2342 (c) 2340 (d) 2347
  23. 23.What is the remainder when 249 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
  24. 24.What is the remainder when 187 is divided by 3? (a) 0 (b) 2 (c) 1
  25. 25.Which of these numbers is divisible by 8? (a) 1905 (b) 1903 (c) 1894 (d) 1896
  26. 26.What is the remainder when 203 is divided by 12? (a) 1 (b) 11 (c) 10 (d) 9
  27. 27.What is the remainder when 222 is divided by 10? (a) 2 (b) 8 (c) 1 (d) 3
  28. 28.What is the remainder when 271 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
  29. 29.Which of these numbers is divisible by 4? (a) 345 (b) 354 (c) 348 (d) 349
  30. 30.What is the remainder when 105 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
  31. 31.What is the smallest number you must add to 204 to make it divisible by 8? (a) 6 (b) 4 (c) 5 (d) 3
  32. 32.Which of these numbers is divisible by 4? (a) 2201 (b) 2204 (c) 2209 (d) 2206
  33. 33.What is the smallest number you must add to 92 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
  34. 34.Which of these numbers is divisible by 3? (a) 1184 (b) 1181 (c) 1179 (d) 1178
  35. 35.What is the units digit of 2^21? (a) 6 (b) 2 (c) 8 (d) 4
  36. 36.Which of these numbers is divisible by 4? (a) 1280 (b) 1285 (c) 1278 (d) 1282
  37. 37.What is the smallest number you must add to 237 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
  38. 38.Which of these numbers is divisible by 6? (a) 3800 (b) 3792 (c) 3790 (d) 3796
  39. 39.What is the units digit of 3^34? (a) 7 (b) 9 (c) 3 (d) 1
  40. 40.What is the smallest number you must add to 277 to make it divisible by 8? (a) 5 (b) 4 (c) 2 (d) 3

Answer key

  1. 9262
  2. 795
  3. 2
  4. 6
  5. 5
  6. 1
  7. 1
  8. 2
  9. 2952
  10. 3
  11. 5
  12. 3
  13. 6
  14. 1
  15. 8
  16. 1984
  17. 7
  18. 5
  19. 1860
  20. 1
  21. 4
  22. 2340
  23. 4
  24. 1
  25. 1896
  26. 11
  27. 2
  28. 3
  29. 348
  30. 1
  31. 4
  32. 2204
  33. 3
  34. 1179
  35. 2
  36. 1280
  37. 3
  38. 3792
  39. 9
  40. 3

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