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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 812 · 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.Which of these numbers is divisible by 11? (a) 3596 (b) 3583 (c) 3599 (d) 3586
  2. 2.What is the remainder when 374 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
  3. 3.What is the remainder when 121 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
  4. 4.Which of these numbers is divisible by 3? (a) 398 (b) 404 (c) 401 (d) 399
  5. 5.What is the remainder when 728 is divided by 12? (a) 7 (b) 8 (c) 9 (d) 4
  6. 6.What is the smallest number you must add to 86 to make it divisible by 3? (a) 2 (b) 1
  7. 7.What is the smallest number you must add to 386 to make it divisible by 8? (a) 7 (b) 5 (c) 2 (d) 6
  8. 8.What is the remainder when 502 is divided by 9? (a) 8 (b) 7 (c) 2 (d) 6
  9. 9.Which of these numbers is divisible by 4? (a) 3084 (b) 3090 (c) 3082 (d) 3086
  10. 10.What is the units digit of 8^53? (a) 4 (b) 2 (c) 6 (d) 8
  11. 11.What is the units digit of 8^22? (a) 8 (b) 2 (c) 6 (d) 4
  12. 12.What is the remainder when 496 is divided by 7? (a) 1 (b) 5 (c) 6 (d) 4
  13. 13.What is the remainder when 862 is divided by 12? (a) 9 (b) 2 (c) 10 (d) 11
  14. 14.What is the remainder when 140 is divided by 3? (a) 2 (b) 0 (c) 1
  15. 15.What is the remainder when 107 is divided by 3? (a) 2 (b) 0 (c) 1
  16. 16.What is the units digit of 4^55? (a) 5 (b) 4 (c) 6 (d) 3
  17. 17.Which of these numbers is divisible by 3? (a) 446 (b) 443 (c) 449 (d) 444
  18. 18.What is the smallest number you must add to 541 to make it divisible by 9? (a) 7 (b) 1 (c) 8 (d) 6
  19. 19.What is the smallest number you must add to 104 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 5
  20. 20.What is the remainder when 492 is divided by 11? (a) 3 (b) 7 (c) 8 (d) 9
  21. 21.What is the remainder when 156 is divided by 11? (a) 9 (b) 2 (c) 1 (d) 3
  22. 22.Which of these numbers is divisible by 6? (a) 1237 (b) 1242 (c) 1252 (d) 1244
  23. 23.Which of these numbers is divisible by 6? (a) 4304 (b) 4291 (c) 4298 (d) 4296
  24. 24.Which of these numbers is divisible by 3? (a) 531 (b) 529 (c) 535 (d) 533
  25. 25.What is the smallest number you must add to 464 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 2
  26. 26.What is the smallest number you must add to 285 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 2
  27. 27.What is the remainder when 205 is divided by 3? (a) 1 (b) 0 (c) 2
  28. 28.What is the remainder when 197 is divided by 3? (a) 0 (b) 2 (c) 1
  29. 29.What is the units digit of 4^20? (a) 7 (b) 4 (c) 5 (d) 6
  30. 30.What is the units digit of 9^38? (a) 2 (b) 0 (c) 1 (d) 9
  31. 31.What is the smallest number you must add to 707 to make it divisible by 12? (a) 3 (b) 1 (c) 11 (d) 2
  32. 32.What is the smallest number you must add to 628 to make it divisible by 12? (a) 7 (b) 9 (c) 4 (d) 8
  33. 33.What is the units digit of 2^56? (a) 8 (b) 4 (c) 2 (d) 6
  34. 34.Which of these numbers is divisible by 9? (a) 7094 (b) 7092 (c) 7104 (d) 7091
  35. 35.What is the smallest number you must add to 91 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  36. 36.What is the units digit of 3^11? (a) 1 (b) 3 (c) 9 (d) 7
  37. 37.What is the smallest number you must add to 445 to make it divisible by 7? (a) 4 (b) 5 (c) 3 (d) 2
  38. 38.What is the units digit of 3^46? (a) 1 (b) 7 (c) 3 (d) 9
  39. 39.What is the units digit of 3^28? (a) 3 (b) 1 (c) 9 (d) 7
  40. 40.What is the remainder when 118 is divided by 3? (a) 0 (b) 1 (c) 2

Answer key

  1. 3586
  2. 4
  3. 1
  4. 399
  5. 8
  6. 1
  7. 6
  8. 7
  9. 3084
  10. 8
  11. 4
  12. 6
  13. 10
  14. 2
  15. 2
  16. 4
  17. 444
  18. 8
  19. 4
  20. 8
  21. 2
  22. 1242
  23. 4296
  24. 531
  25. 4
  26. 3
  27. 1
  28. 2
  29. 6
  30. 1
  31. 1
  32. 8
  33. 6
  34. 7092
  35. 1
  36. 7
  37. 3
  38. 9
  39. 1
  40. 1

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