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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 136 · 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 634 to make it divisible by 12? (a) 3 (b) 2 (c) 10 (d) 1
  2. 2.What is the smallest number you must add to 229 to make it divisible by 7? (a) 2 (b) 5 (c) 1 (d) 3
  3. 3.What is the smallest number you must add to 269 to make it divisible by 10? (a) 2 (b) 3 (c) 1 (d) 9
  4. 4.What is the smallest number you must add to 86 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
  5. 5.What is the units digit of 3^44? (a) 9 (b) 7 (c) 1 (d) 3
  6. 6.What is the units digit of 2^12? (a) 4 (b) 2 (c) 8 (d) 6
  7. 7.What is the remainder when 113 is divided by 6? (a) 4 (b) 1 (c) 5 (d) 3
  8. 8.What is the remainder when 299 is divided by 9? (a) 7 (b) 3 (c) 1 (d) 2
  9. 9.Which of these numbers is divisible by 3? (a) 1636 (b) 1634 (c) 1640 (d) 1635
  10. 10.Which of these numbers is divisible by 4? (a) 2812 (b) 2817 (c) 2809 (d) 2815
  11. 11.What is the smallest number you must add to 195 to make it divisible by 7? (a) 1 (b) 2 (c) 6 (d) 3
  12. 12.What is the smallest number you must add to 308 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  13. 13.What is the units digit of 8^22? (a) 6 (b) 8 (c) 2 (d) 4
  14. 14.What is the remainder when 471 is divided by 11? (a) 10 (b) 8 (c) 2 (d) 9
  15. 15.What is the units digit of 2^26? (a) 2 (b) 6 (c) 4 (d) 8
  16. 16.What is the smallest number you must add to 116 to make it divisible by 11? (a) 6 (b) 7 (c) 4 (d) 5
  17. 17.What is the units digit of 2^50? (a) 4 (b) 8 (c) 2 (d) 6
  18. 18.Which of these numbers is divisible by 6? (a) 3050 (b) 3057 (c) 3061 (d) 3054
  19. 19.What is the remainder when 43 is divided by 3? (a) 0 (b) 2 (c) 1
  20. 20.What is the remainder when 443 is divided by 7? (a) 2 (b) 3 (c) 5 (d) 1
  21. 21.What is the remainder when 328 is divided by 11? (a) 2 (b) 10 (c) 9 (d) 8
  22. 22.What is the smallest number you must add to 313 to make it divisible by 10? (a) 3 (b) 8 (c) 7 (d) 6
  23. 23.What is the remainder when 292 is divided by 8? (a) 6 (b) 3 (c) 5 (d) 4
  24. 24.What is the smallest number you must add to 478 to make it divisible by 9? (a) 7 (b) 1 (c) 6 (d) 8
  25. 25.Which of these numbers is divisible by 4? (a) 305 (b) 303 (c) 311 (d) 304
  26. 26.Which of these numbers is divisible by 4? (a) 3325 (b) 3320 (c) 3322 (d) 3317
  27. 27.What is the remainder when 511 is divided by 12? (a) 5 (b) 7 (c) 8 (d) 6
  28. 28.What is the smallest number you must add to 356 to make it divisible by 12? (a) 5 (b) 8 (c) 3 (d) 4
  29. 29.What is the remainder when 136 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
  30. 30.Which of these numbers is divisible by 4? (a) 578 (b) 570 (c) 573 (d) 572
  31. 31.Which of these numbers is divisible by 9? (a) 4147 (b) 4136 (c) 4140 (d) 4156
  32. 32.What is the units digit of 9^29? (a) 7 (b) 1 (c) 8 (d) 9
  33. 33.What is the smallest number you must add to 236 to make it divisible by 8? (a) 6 (b) 3 (c) 5 (d) 4
  34. 34.What is the units digit of 8^36? (a) 2 (b) 8 (c) 6 (d) 4
  35. 35.What is the remainder when 413 is divided by 6? (a) 5 (b) 1 (c) 4 (d) 3
  36. 36.What is the remainder when 173 is divided by 3? (a) 0 (b) 2 (c) 1
  37. 37.Which of these numbers is divisible by 9? (a) 4062 (b) 4068 (c) 4075 (d) 4084
  38. 38.What is the smallest number you must add to 97 to make it divisible by 3? (a) 2 (b) 1
  39. 39.Which of these numbers is divisible by 6? (a) 4606 (b) 4608 (c) 4609 (d) 4617
  40. 40.Which of these numbers is divisible by 8? (a) 374 (b) 366 (c) 358 (d) 360

Answer key

  1. 2
  2. 2
  3. 1
  4. 4
  5. 1
  6. 6
  7. 5
  8. 2
  9. 1635
  10. 2812
  11. 1
  12. 2
  13. 4
  14. 9
  15. 4
  16. 5
  17. 4
  18. 3054
  19. 1
  20. 2
  21. 9
  22. 7
  23. 4
  24. 8
  25. 304
  26. 3320
  27. 7
  28. 4
  29. 1
  30. 572
  31. 4140
  32. 9
  33. 4
  34. 6
  35. 5
  36. 2
  37. 4068
  38. 2
  39. 4608
  40. 360

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