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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 434 · 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 487 is divided by 9? (a) 8 (b) 2 (c) 0 (d) 1
  2. 2.What is the remainder when 577 is divided by 9? (a) 8 (b) 1 (c) 2 (d) 0
  3. 3.What is the units digit of 9^59? (a) 8 (b) 9 (c) 7 (d) 1
  4. 4.Which of these numbers is divisible by 6? (a) 3919 (b) 3918 (c) 3929 (d) 3913
  5. 5.What is the smallest number you must add to 615 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 7
  6. 6.Which of these numbers is divisible by 11? (a) 9677 (b) 9669 (c) 9664 (d) 9684
  7. 7.What is the smallest number you must add to 125 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  8. 8.What is the smallest number you must add to 333 to make it divisible by 8? (a) 4 (b) 3 (c) 2 (d) 5
  9. 9.Which of these numbers is divisible by 9? (a) 1660 (b) 1649 (c) 1646 (d) 1647
  10. 10.What is the remainder when 225 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
  11. 11.What is the smallest number you must add to 773 to make it divisible by 12? (a) 6 (b) 5 (c) 8 (d) 7
  12. 12.Which of these numbers is divisible by 11? (a) 3339 (b) 3352 (c) 3327 (d) 3333
  13. 13.What is the remainder when 98 is divided by 3? (a) 1 (b) 2 (c) 0
  14. 14.Which of these numbers is divisible by 3? (a) 1958 (b) 1964 (c) 1959 (d) 1960
  15. 15.Which of these numbers is divisible by 3? (a) 2568 (b) 2570 (c) 2572 (d) 2566
  16. 16.What is the smallest number you must add to 967 to make it divisible by 12? (a) 4 (b) 5 (c) 7 (d) 6
  17. 17.Which of these numbers is divisible by 4? (a) 2657 (b) 2652 (c) 2653 (d) 2649
  18. 18.What is the smallest number you must add to 331 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  19. 19.Which of these numbers is divisible by 9? (a) 2406 (b) 2391 (c) 2399 (d) 2394
  20. 20.What is the smallest number you must add to 362 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  21. 21.What is the remainder when 254 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
  22. 22.What is the remainder when 708 is divided by 10? (a) 2 (b) 8 (c) 7 (d) 9
  23. 23.What is the smallest number you must add to 290 to make it divisible by 9? (a) 8 (b) 6 (c) 7 (d) 2
  24. 24.What is the units digit of 4^27? (a) 5 (b) 3 (c) 6 (d) 4
  25. 25.What is the remainder when 128 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
  26. 26.What is the smallest number you must add to 561 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 1
  27. 27.What is the smallest number you must add to 391 to make it divisible by 8? (a) 7 (b) 2 (c) 3 (d) 1
  28. 28.What is the smallest number you must add to 815 to make it divisible by 12? (a) 11 (b) 2 (c) 3 (d) 1
  29. 29.What is the units digit of 9^36? (a) 2 (b) 1 (c) 0 (d) 9
  30. 30.Which of these numbers is divisible by 6? (a) 1301 (b) 1302 (c) 1304 (d) 1313
  31. 31.What is the units digit of 4^29? (a) 4 (b) 3 (c) 5 (d) 6
  32. 32.What is the smallest number you must add to 53 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
  33. 33.What is the smallest number you must add to 565 to make it divisible by 9? (a) 7 (b) 1 (c) 3 (d) 2
  34. 34.What is the units digit of 8^16? (a) 8 (b) 6 (c) 4 (d) 2
  35. 35.What is the smallest number you must add to 204 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
  36. 36.What is the smallest number you must add to 115 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  37. 37.What is the units digit of 7^39? (a) 9 (b) 1 (c) 3 (d) 7
  38. 38.What is the units digit of 2^57? (a) 8 (b) 4 (c) 6 (d) 2
  39. 39.What is the units digit of 9^29? (a) 9 (b) 8 (c) 1 (d) 7
  40. 40.What is the smallest number you must add to 345 to make it divisible by 10? (a) 6 (b) 5 (c) 7 (d) 4

Answer key

  1. 1
  2. 1
  3. 9
  4. 3918
  5. 6
  6. 9669
  7. 3
  8. 3
  9. 1647
  10. 1
  11. 7
  12. 3333
  13. 2
  14. 1959
  15. 2568
  16. 5
  17. 2652
  18. 4
  19. 2394
  20. 3
  21. 4
  22. 8
  23. 7
  24. 4
  25. 3
  26. 7
  27. 1
  28. 1
  29. 1
  30. 1302
  31. 4
  32. 2
  33. 2
  34. 6
  35. 4
  36. 1
  37. 3
  38. 2
  39. 9
  40. 5

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