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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 60 · 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 158 to make it divisible by 7? (a) 4 (b) 5 (c) 3 (d) 2
  2. 2.Which of these numbers is divisible by 8? (a) 1999 (b) 1984 (c) 1989 (d) 1977
  3. 3.What is the units digit of 2^59? (a) 8 (b) 4 (c) 6 (d) 2
  4. 4.What is the smallest number you must add to 282 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  5. 5.What is the smallest number you must add to 241 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  6. 6.What is the smallest number you must add to 526 to make it divisible by 9? (a) 4 (b) 6 (c) 5 (d) 7
  7. 7.What is the remainder when 275 is divided by 8? (a) 2 (b) 4 (c) 5 (d) 3
  8. 8.What is the units digit of 4^18? (a) 7 (b) 4 (c) 5 (d) 6
  9. 9.What is the remainder when 533 is divided by 11? (a) 6 (b) 7 (c) 4 (d) 5
  10. 10.Which of these numbers is divisible by 6? (a) 3994 (b) 3990 (c) 3989 (d) 3997
  11. 11.What is the remainder when 305 is divided by 11? (a) 9 (b) 3 (c) 8 (d) 7
  12. 12.What is the remainder when 542 is divided by 10? (a) 3 (b) 1 (c) 8 (d) 2
  13. 13.What is the remainder when 37 is divided by 3? (a) 2 (b) 1 (c) 0
  14. 14.What is the remainder when 428 is divided by 7? (a) 2 (b) 1 (c) 6 (d) 0
  15. 15.What is the remainder when 83 is divided by 3? (a) 1 (b) 0 (c) 2
  16. 16.What is the remainder when 166 is divided by 7? (a) 2 (b) 4 (c) 6 (d) 5
  17. 17.What is the remainder when 170 is divided by 8? (a) 6 (b) 3 (c) 2 (d) 1
  18. 18.What is the smallest number you must add to 124 to make it divisible by 9? (a) 1 (b) 2 (c) 3 (d) 7
  19. 19.What is the remainder when 103 is divided by 9? (a) 4 (b) 6 (c) 3 (d) 5
  20. 20.What is the units digit of 4^22? (a) 7 (b) 5 (c) 6 (d) 4
  21. 21.What is the remainder when 550 is divided by 12? (a) 9 (b) 11 (c) 10 (d) 2
  22. 22.What is the smallest number you must add to 342 to make it divisible by 7? (a) 2 (b) 1 (c) 6 (d) 3
  23. 23.What is the units digit of 7^44? (a) 3 (b) 9 (c) 7 (d) 1
  24. 24.What is the remainder when 285 is divided by 12? (a) 10 (b) 3 (c) 9 (d) 8
  25. 25.What is the remainder when 185 is divided by 8? (a) 0 (b) 1 (c) 2 (d) 7
  26. 26.What is the smallest number you must add to 429 to make it divisible by 7? (a) 2 (b) 5 (c) 4 (d) 6
  27. 27.What is the remainder when 53 is divided by 3? (a) 0 (b) 1 (c) 2
  28. 28.What is the smallest number you must add to 124 to make it divisible by 10? (a) 7 (b) 5 (c) 4 (d) 6
  29. 29.What is the smallest number you must add to 149 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  30. 30.Which of these numbers is divisible by 11? (a) 4061 (b) 4091 (c) 4070 (d) 4076
  31. 31.What is the remainder when 169 is divided by 3? (a) 0 (b) 2 (c) 1
  32. 32.What is the units digit of 3^20? (a) 1 (b) 3 (c) 9 (d) 7
  33. 33.What is the smallest number you must add to 368 to make it divisible by 9? (a) 2 (b) 1 (c) 8 (d) 3
  34. 34.What is the smallest number you must add to 802 to make it divisible by 10? (a) 2 (b) 9 (c) 8 (d) 7
  35. 35.Which of these numbers is divisible by 3? (a) 1036 (b) 1043 (c) 1038 (d) 1040
  36. 36.What is the smallest number you must add to 398 to make it divisible by 8? (a) 6 (b) 3 (c) 2 (d) 1
  37. 37.Which of these numbers is divisible by 6? (a) 2581 (b) 2574 (c) 2579 (d) 2570
  38. 38.Which of these numbers is divisible by 4? (a) 2748 (b) 2751 (c) 2745 (d) 2755
  39. 39.What is the smallest number you must add to 207 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  40. 40.What is the smallest number you must add to 267 to make it divisible by 7? (a) 4 (b) 6 (c) 1 (d) 5

Answer key

  1. 3
  2. 1984
  3. 8
  4. 2
  5. 3
  6. 5
  7. 3
  8. 6
  9. 5
  10. 3990
  11. 8
  12. 2
  13. 1
  14. 1
  15. 2
  16. 5
  17. 2
  18. 2
  19. 4
  20. 6
  21. 10
  22. 1
  23. 1
  24. 9
  25. 1
  26. 5
  27. 2
  28. 6
  29. 1
  30. 4070
  31. 1
  32. 1
  33. 1
  34. 8
  35. 1038
  36. 2
  37. 2574
  38. 2748
  39. 1
  40. 6

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