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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 615 · 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 184 is divided by 7? (a) 5 (b) 2 (c) 1 (d) 3
  2. 2.Which of these numbers is divisible by 6? (a) 5238 (b) 5235 (c) 5246 (d) 5240
  3. 3.What is the smallest number you must add to 339 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  4. 4.Which of these numbers is divisible by 11? (a) 4994 (b) 5003 (c) 4984 (d) 5010
  5. 5.What is the smallest number you must add to 227 to make it divisible by 7? (a) 5 (b) 3 (c) 6 (d) 4
  6. 6.What is the units digit of 4^26? (a) 4 (b) 6 (c) 7 (d) 5
  7. 7.What is the units digit of 9^45? (a) 1 (b) 8 (c) 7 (d) 9
  8. 8.Which of these numbers is divisible by 4? (a) 2350 (b) 2344 (c) 2346 (d) 2343
  9. 9.What is the smallest number you must add to 215 to make it divisible by 7? (a) 1 (b) 2 (c) 3 (d) 5
  10. 10.What is the smallest number you must add to 123 to make it divisible by 7? (a) 5 (b) 2 (c) 4 (d) 3
  11. 11.What is the remainder when 197 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 1
  12. 12.What is the units digit of 3^47? (a) 1 (b) 7 (c) 3 (d) 9
  13. 13.What is the smallest number you must add to 160 to make it divisible by 7? (a) 3 (b) 2 (c) 6 (d) 1
  14. 14.What is the units digit of 4^52? (a) 6 (b) 7 (c) 4 (d) 5
  15. 15.What is the units digit of 8^37? (a) 6 (b) 4 (c) 2 (d) 8
  16. 16.What is the smallest number you must add to 53 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  17. 17.What is the remainder when 238 is divided by 8? (a) 7 (b) 5 (c) 2 (d) 6
  18. 18.What is the smallest number you must add to 956 to make it divisible by 12? (a) 4 (b) 8 (c) 3 (d) 5
  19. 19.What is the remainder when 62 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  20. 20.What is the remainder when 135 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 2
  21. 21.What is the smallest number you must add to 399 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 7
  22. 22.Which of these numbers is divisible by 4? (a) 1271 (b) 1275 (c) 1277 (d) 1272
  23. 23.Which of these numbers is divisible by 8? (a) 3653 (b) 3660 (c) 3647 (d) 3648
  24. 24.What is the smallest number you must add to 40 to make it divisible by 3? (a) 1 (b) 2
  25. 25.What is the smallest number you must add to 309 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
  26. 26.What is the smallest number you must add to 430 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
  27. 27.What is the remainder when 402 is divided by 10? (a) 1 (b) 2 (c) 8 (d) 3
  28. 28.What is the remainder when 359 is divided by 9? (a) 6 (b) 1 (c) 8 (d) 7
  29. 29.What is the remainder when 242 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
  30. 30.Which of these numbers is divisible by 8? (a) 1362 (b) 1373 (c) 1360 (d) 1357
  31. 31.What is the remainder when 379 is divided by 7? (a) 0 (b) 1 (c) 6 (d) 2
  32. 32.What is the units digit of 8^25? (a) 2 (b) 6 (c) 8 (d) 4
  33. 33.What is the units digit of 7^52? (a) 9 (b) 3 (c) 7 (d) 1
  34. 34.What is the units digit of 9^18? (a) 1 (b) 0 (c) 9 (d) 2
  35. 35.What is the units digit of 4^20? (a) 4 (b) 6 (c) 7 (d) 5
  36. 36.What is the remainder when 427 is divided by 9? (a) 4 (b) 3 (c) 6 (d) 5
  37. 37.What is the remainder when 554 is divided by 7? (a) 6 (b) 2 (c) 1 (d) 0
  38. 38.What is the smallest number you must add to 247 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 7
  39. 39.What is the units digit of 9^25? (a) 8 (b) 9 (c) 7 (d) 1
  40. 40.What is the units digit of 2^21? (a) 8 (b) 6 (c) 4 (d) 2

Answer key

  1. 2
  2. 5238
  3. 1
  4. 4994
  5. 4
  6. 6
  7. 9
  8. 2344
  9. 2
  10. 3
  11. 8
  12. 7
  13. 1
  14. 6
  15. 8
  16. 3
  17. 6
  18. 4
  19. 2
  20. 3
  21. 6
  22. 1272
  23. 3648
  24. 2
  25. 3
  26. 2
  27. 2
  28. 8
  29. 2
  30. 1360
  31. 1
  32. 8
  33. 1
  34. 1
  35. 6
  36. 4
  37. 1
  38. 1
  39. 9
  40. 2

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