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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 231 · 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 9? (a) 2205 (b) 2199 (c) 2210 (d) 2220
  2. 2.Which of these numbers is divisible by 4? (a) 832 (b) 831 (c) 839 (d) 833
  3. 3.Which of these numbers is divisible by 4? (a) 2010 (b) 2007 (c) 2015 (d) 2008
  4. 4.What is the remainder when 349 is divided by 9? (a) 2 (b) 8 (c) 7 (d) 6
  5. 5.Which of these numbers is divisible by 9? (a) 5954 (b) 5942 (c) 5963 (d) 5949
  6. 6.Which of these numbers is divisible by 9? (a) 1601 (b) 1605 (c) 1602 (d) 1612
  7. 7.Which of these numbers is divisible by 11? (a) 7029 (b) 7026 (c) 7043 (d) 7038
  8. 8.What is the units digit of 2^51? (a) 8 (b) 4 (c) 6 (d) 2
  9. 9.What is the remainder when 209 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
  10. 10.What is the smallest number you must add to 303 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  11. 11.What is the units digit of 4^17? (a) 4 (b) 3 (c) 5 (d) 6
  12. 12.What is the smallest number you must add to 119 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  13. 13.Which of these numbers is divisible by 4? (a) 2269 (b) 2264 (c) 2261 (d) 2267
  14. 14.What is the remainder when 154 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
  15. 15.What is the smallest number you must add to 151 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  16. 16.What is the smallest number you must add to 65 to make it divisible by 3? (a) 2 (b) 1
  17. 17.What is the remainder when 406 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 5
  18. 18.What is the smallest number you must add to 218 to make it divisible by 10? (a) 3 (b) 8 (c) 1 (d) 2
  19. 19.What is the remainder when 59 is divided by 3? (a) 2 (b) 1 (c) 0
  20. 20.What is the remainder when 630 is divided by 8? (a) 7 (b) 5 (c) 6 (d) 2
  21. 21.Which of these numbers is divisible by 11? (a) 4938 (b) 4914 (c) 4917 (d) 4922
  22. 22.Which of these numbers is divisible by 9? (a) 3665 (b) 3662 (c) 3654 (d) 3651
  23. 23.Which of these numbers is divisible by 6? (a) 968 (b) 966 (c) 962 (d) 973
  24. 24.What is the smallest number you must add to 789 to make it divisible by 10? (a) 9 (b) 2 (c) 3 (d) 1
  25. 25.What is the smallest number you must add to 453 to make it divisible by 7? (a) 5 (b) 1 (c) 3 (d) 2
  26. 26.Which of these numbers is divisible by 4? (a) 242 (b) 238 (c) 233 (d) 236
  27. 27.What is the units digit of 3^40? (a) 9 (b) 7 (c) 1 (d) 3
  28. 28.What is the remainder when 65 is divided by 3? (a) 2 (b) 1 (c) 0
  29. 29.What is the smallest number you must add to 565 to make it divisible by 10? (a) 6 (b) 7 (c) 5 (d) 4
  30. 30.What is the smallest number you must add to 183 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  31. 31.What is the smallest number you must add to 956 to make it divisible by 12? (a) 4 (b) 5 (c) 8 (d) 3
  32. 32.Which of these numbers is divisible by 3? (a) 230 (b) 231 (c) 232 (d) 236
  33. 33.What is the units digit of 3^39? (a) 9 (b) 1 (c) 7 (d) 3
  34. 34.What is the smallest number you must add to 59 to make it divisible by 3? (a) 2 (b) 1
  35. 35.Which of these numbers is divisible by 3? (a) 2297 (b) 2293 (c) 2292 (d) 2290
  36. 36.Which of these numbers is divisible by 11? (a) 2343 (b) 2337 (c) 2346 (d) 2363
  37. 37.What is the units digit of 8^41? (a) 2 (b) 6 (c) 4 (d) 8
  38. 38.What is the smallest number you must add to 229 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
  39. 39.Which of these numbers is divisible by 9? (a) 7622 (b) 7629 (c) 7610 (d) 7614
  40. 40.What is the units digit of 7^28? (a) 1 (b) 9 (c) 3 (d) 7

Answer key

  1. 2205
  2. 832
  3. 2008
  4. 7
  5. 5949
  6. 1602
  7. 7029
  8. 8
  9. 1
  10. 1
  11. 4
  12. 1
  13. 2264
  14. 4
  15. 1
  16. 1
  17. 4
  18. 2
  19. 2
  20. 6
  21. 4917
  22. 3654
  23. 966
  24. 1
  25. 2
  26. 236
  27. 1
  28. 2
  29. 5
  30. 1
  31. 4
  32. 231
  33. 7
  34. 1
  35. 2292
  36. 2343
  37. 8
  38. 5
  39. 7614
  40. 1

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