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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 284 · 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 109 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
  2. 2.What is the smallest number you must add to 774 to make it divisible by 11? (a) 6 (b) 4 (c) 8 (d) 7
  3. 3.What is the remainder when 491 is divided by 7? (a) 1 (b) 0 (c) 2 (d) 6
  4. 4.Which of these numbers is divisible by 9? (a) 3195 (b) 3209 (c) 3193 (d) 3201
  5. 5.What is the smallest number you must add to 65 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  6. 6.What is the units digit of 7^33? (a) 9 (b) 3 (c) 1 (d) 7
  7. 7.What is the smallest number you must add to 303 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 2
  8. 8.Which of these numbers is divisible by 11? (a) 5260 (b) 5251 (c) 5276 (d) 5258
  9. 9.What is the remainder when 707 is divided by 9? (a) 4 (b) 7 (c) 6 (d) 5
  10. 10.What is the units digit of 7^41? (a) 3 (b) 9 (c) 1 (d) 7
  11. 11.What is the smallest number you must add to 626 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 2
  12. 12.What is the remainder when 59 is divided by 3? (a) 0 (b) 2 (c) 1
  13. 13.What is the smallest number you must add to 283 to make it divisible by 9? (a) 7 (b) 4 (c) 6 (d) 5
  14. 14.What is the units digit of 7^17? (a) 1 (b) 7 (c) 3 (d) 9
  15. 15.What is the units digit of 2^60? (a) 2 (b) 8 (c) 6 (d) 4
  16. 16.What is the smallest number you must add to 357 to make it divisible by 8? (a) 5 (b) 2 (c) 3 (d) 4
  17. 17.What is the units digit of 3^37? (a) 3 (b) 7 (c) 9 (d) 1
  18. 18.What is the smallest number you must add to 103 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
  19. 19.What is the smallest number you must add to 217 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  20. 20.What is the remainder when 344 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
  21. 21.What is the remainder when 57 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
  22. 22.What is the units digit of 3^19? (a) 1 (b) 9 (c) 7 (d) 3
  23. 23.What is the remainder when 229 is divided by 9? (a) 6 (b) 4 (c) 5 (d) 3
  24. 24.What is the smallest number you must add to 303 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
  25. 25.What is the smallest number you must add to 513 to make it divisible by 8? (a) 1 (b) 6 (c) 7 (d) 5
  26. 26.What is the smallest number you must add to 859 to make it divisible by 12? (a) 4 (b) 7 (c) 5 (d) 6
  27. 27.What is the remainder when 64 is divided by 3? (a) 2 (b) 0 (c) 1
  28. 28.What is the smallest number you must add to 320 to make it divisible by 12? (a) 3 (b) 8 (c) 5 (d) 4
  29. 29.Which of these numbers is divisible by 8? (a) 5712 (b) 5713 (c) 5721 (d) 5708
  30. 30.What is the units digit of 8^25? (a) 4 (b) 2 (c) 8 (d) 6
  31. 31.What is the remainder when 365 is divided by 6? (a) 3 (b) 5 (c) 1 (d) 4
  32. 32.What is the units digit of 4^12? (a) 6 (b) 4 (c) 7 (d) 5
  33. 33.Which of these numbers is divisible by 3? (a) 1151 (b) 1153 (c) 1152 (d) 1157
  34. 34.What is the smallest number you must add to 274 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  35. 35.What is the smallest number you must add to 358 to make it divisible by 7? (a) 5 (b) 1 (c) 6 (d) 4
  36. 36.What is the remainder when 577 is divided by 8? (a) 7 (b) 0 (c) 2 (d) 1
  37. 37.What is the smallest number you must add to 121 to make it divisible by 9? (a) 5 (b) 7 (c) 4 (d) 6
  38. 38.What is the remainder when 262 is divided by 6? (a) 5 (b) 3 (c) 2 (d) 4
  39. 39.What is the remainder when 310 is divided by 11? (a) 3 (b) 9 (c) 1 (d) 2
  40. 40.What is the smallest number you must add to 784 to make it divisible by 10? (a) 5 (b) 4 (c) 7 (d) 6

Answer key

  1. 5
  2. 7
  3. 1
  4. 3195
  5. 3
  6. 7
  7. 5
  8. 5258
  9. 5
  10. 7
  11. 6
  12. 2
  13. 5
  14. 7
  15. 6
  16. 3
  17. 3
  18. 2
  19. 3
  20. 4
  21. 2
  22. 7
  23. 4
  24. 2
  25. 7
  26. 5
  27. 1
  28. 4
  29. 5712
  30. 8
  31. 5
  32. 6
  33. 1152
  34. 2
  35. 6
  36. 1
  37. 5
  38. 4
  39. 2
  40. 6

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