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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 166 · 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 562 is divided by 7? (a) 3 (b) 1 (c) 2 (d) 5
  2. 2.What is the smallest number you must add to 829 to make it divisible by 12? (a) 10 (b) 9 (c) 11 (d) 1
  3. 3.What is the smallest number you must add to 650 to make it divisible by 11? (a) 8 (b) 1 (c) 10 (d) 9
  4. 4.What is the smallest number you must add to 182 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  5. 5.What is the smallest number you must add to 102 to make it divisible by 10? (a) 2 (b) 7 (c) 9 (d) 8
  6. 6.What is the units digit of 7^15? (a) 7 (b) 9 (c) 3 (d) 1
  7. 7.What is the units digit of 9^28? (a) 9 (b) 0 (c) 1 (d) 2
  8. 8.What is the units digit of 7^22? (a) 9 (b) 7 (c) 3 (d) 1
  9. 9.What is the units digit of 7^39? (a) 3 (b) 7 (c) 9 (d) 1
  10. 10.What is the smallest number you must add to 73 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  11. 11.What is the units digit of 9^15? (a) 1 (b) 9 (c) 7 (d) 8
  12. 12.Which of these numbers is divisible by 11? (a) 1095 (b) 1102 (c) 1088 (d) 1089
  13. 13.What is the remainder when 105 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 2
  14. 14.What is the units digit of 4^17? (a) 5 (b) 4 (c) 6 (d) 3
  15. 15.What is the smallest number you must add to 459 to make it divisible by 10? (a) 1 (b) 9 (c) 2 (d) 3
  16. 16.What is the units digit of 3^12? (a) 3 (b) 9 (c) 7 (d) 1
  17. 17.What is the remainder when 677 is divided by 10? (a) 6 (b) 7 (c) 3 (d) 8
  18. 18.What is the units digit of 4^18? (a) 5 (b) 7 (c) 6 (d) 4
  19. 19.What is the smallest number you must add to 94 to make it divisible by 7? (a) 3 (b) 5 (c) 4 (d) 6
  20. 20.What is the units digit of 7^31? (a) 9 (b) 1 (c) 7 (d) 3
  21. 21.What is the units digit of 9^49? (a) 7 (b) 8 (c) 1 (d) 9
  22. 22.What is the remainder when 249 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
  23. 23.What is the units digit of 2^37? (a) 4 (b) 6 (c) 8 (d) 2
  24. 24.What is the smallest number you must add to 391 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
  25. 25.What is the smallest number you must add to 122 to make it divisible by 12? (a) 11 (b) 2 (c) 10 (d) 9
  26. 26.What is the smallest number you must add to 498 to make it divisible by 11? (a) 9 (b) 8 (c) 3 (d) 7
  27. 27.Which of these numbers is divisible by 6? (a) 1997 (b) 1981 (c) 1990 (d) 1986
  28. 28.What is the smallest number you must add to 369 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 5
  29. 29.What is the units digit of 2^13? (a) 8 (b) 6 (c) 2 (d) 4
  30. 30.What is the remainder when 289 is divided by 10? (a) 9 (b) 8 (c) 1 (d) 7
  31. 31.What is the remainder when 119 is divided by 3? (a) 0 (b) 1 (c) 2
  32. 32.What is the units digit of 2^24? (a) 4 (b) 8 (c) 2 (d) 6
  33. 33.Which of these numbers is divisible by 9? (a) 1062 (b) 1054 (c) 1077 (d) 1068
  34. 34.What is the remainder when 128 is divided by 6? (a) 1 (b) 2 (c) 3 (d) 4
  35. 35.Which of these numbers is divisible by 3? (a) 2329 (b) 2327 (c) 2324 (d) 2325
  36. 36.What is the remainder when 662 is divided by 12? (a) 2 (b) 3 (c) 1 (d) 10
  37. 37.What is the units digit of 2^40? (a) 2 (b) 6 (c) 8 (d) 4
  38. 38.What is the remainder when 369 is divided by 11? (a) 6 (b) 7 (c) 5 (d) 8
  39. 39.What is the smallest number you must add to 242 to make it divisible by 9? (a) 1 (b) 2 (c) 8 (d) 3
  40. 40.What is the smallest number you must add to 155 to make it divisible by 3? (a) 1 (b) 2

Answer key

  1. 2
  2. 11
  3. 10
  4. 2
  5. 8
  6. 3
  7. 1
  8. 9
  9. 3
  10. 3
  11. 9
  12. 1089
  13. 3
  14. 4
  15. 1
  16. 1
  17. 7
  18. 6
  19. 4
  20. 3
  21. 9
  22. 1
  23. 2
  24. 4
  25. 10
  26. 8
  27. 1986
  28. 2
  29. 2
  30. 9
  31. 2
  32. 6
  33. 1062
  34. 2
  35. 2325
  36. 2
  37. 6
  38. 6
  39. 1
  40. 1

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