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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 73 · 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 50 is divided by 3? (a) 0 (b) 1 (c) 2
  2. 2.Which of these numbers is divisible by 4? (a) 2456 (b) 2457 (c) 2454 (d) 2462
  3. 3.Which of these numbers is divisible by 6? (a) 3387 (b) 3379 (c) 3384 (d) 3391
  4. 4.What is the smallest number you must add to 390 to make it divisible by 8? (a) 3 (b) 1 (c) 6 (d) 2
  5. 5.What is the remainder when 586 is divided by 9? (a) 1 (b) 8 (c) 0 (d) 2
  6. 6.What is the units digit of 9^35? (a) 9 (b) 7 (c) 8 (d) 1
  7. 7.Which of these numbers is divisible by 11? (a) 2129 (b) 2151 (c) 2139 (d) 2134
  8. 8.What is the units digit of 3^36? (a) 7 (b) 3 (c) 1 (d) 9
  9. 9.Which of these numbers is divisible by 3? (a) 1013 (b) 1015 (c) 1018 (d) 1014
  10. 10.What is the smallest number you must add to 846 to make it divisible by 11? (a) 10 (b) 1 (c) 3 (d) 2
  11. 11.What is the smallest number you must add to 185 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  12. 12.What is the smallest number you must add to 246 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  13. 13.What is the remainder when 82 is divided by 3? (a) 0 (b) 1 (c) 2
  14. 14.What is the smallest number you must add to 231 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
  15. 15.What is the smallest number you must add to 596 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 6
  16. 16.Which of these numbers is divisible by 4? (a) 3541 (b) 3545 (c) 3540 (d) 3539
  17. 17.What is the remainder when 724 is divided by 11? (a) 2 (b) 8 (c) 9 (d) 10
  18. 18.Which of these numbers is divisible by 3? (a) 2100 (b) 2101 (c) 2104 (d) 2099
  19. 19.What is the smallest number you must add to 857 to make it divisible by 12? (a) 7 (b) 6 (c) 5 (d) 8
  20. 20.What is the smallest number you must add to 151 to make it divisible by 8? (a) 2 (b) 7 (c) 3 (d) 1
  21. 21.What is the remainder when 306 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
  22. 22.What is the units digit of 4^60? (a) 7 (b) 5 (c) 6 (d) 4
  23. 23.What is the remainder when 145 is divided by 8? (a) 0 (b) 1 (c) 2 (d) 7
  24. 24.Which of these numbers is divisible by 8? (a) 6592 (b) 6588 (c) 6594 (d) 6603
  25. 25.What is the smallest number you must add to 127 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 4
  26. 26.What is the smallest number you must add to 632 to make it divisible by 9? (a) 2 (b) 7 (c) 8 (d) 6
  27. 27.What is the smallest number you must add to 489 to make it divisible by 10? (a) 2 (b) 9 (c) 3 (d) 1
  28. 28.What is the units digit of 2^10? (a) 4 (b) 2 (c) 6 (d) 8
  29. 29.What is the units digit of 8^16? (a) 8 (b) 4 (c) 6 (d) 2
  30. 30.What is the remainder when 79 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
  31. 31.What is the units digit of 7^49? (a) 9 (b) 1 (c) 7 (d) 3
  32. 32.What is the remainder when 67 is divided by 6? (a) 5 (b) 0 (c) 1 (d) 2
  33. 33.What is the smallest number you must add to 71 to make it divisible by 3? (a) 1 (b) 2
  34. 34.Which of these numbers is divisible by 8? (a) 2403 (b) 2415 (c) 2400 (d) 2395
  35. 35.What is the smallest number you must add to 269 to make it divisible by 6? (a) 1 (b) 5 (c) 2 (d) 3
  36. 36.What is the remainder when 73 is divided by 3? (a) 2 (b) 0 (c) 1
  37. 37.What is the remainder when 139 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
  38. 38.What is the units digit of 8^36? (a) 8 (b) 2 (c) 4 (d) 6
  39. 39.What is the units digit of 3^19? (a) 9 (b) 3 (c) 1 (d) 7
  40. 40.What is the units digit of 3^42? (a) 3 (b) 9 (c) 1 (d) 7

Answer key

  1. 2
  2. 2456
  3. 3384
  4. 2
  5. 1
  6. 9
  7. 2134
  8. 1
  9. 1014
  10. 1
  11. 3
  12. 4
  13. 1
  14. 4
  15. 4
  16. 3540
  17. 9
  18. 2100
  19. 7
  20. 1
  21. 2
  22. 6
  23. 1
  24. 6592
  25. 5
  26. 7
  27. 1
  28. 4
  29. 6
  30. 3
  31. 7
  32. 1
  33. 1
  34. 2400
  35. 1
  36. 1
  37. 3
  38. 6
  39. 7
  40. 9

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