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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 890 · 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 9? (a) 742 (b) 732 (c) 726 (d) 729
  2. 2.What is the remainder when 134 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
  3. 3.What is the remainder when 412 is divided by 11? (a) 5 (b) 6 (c) 7 (d) 4
  4. 4.What is the smallest number you must add to 82 to make it divisible by 3? (a) 1 (b) 2
  5. 5.What is the units digit of 9^18? (a) 1 (b) 9 (c) 0 (d) 2
  6. 6.What is the smallest number you must add to 58 to make it divisible by 3? (a) 1 (b) 2
  7. 7.What is the smallest number you must add to 104 to make it divisible by 7? (a) 3 (b) 2 (c) 6 (d) 1
  8. 8.What is the smallest number you must add to 164 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
  9. 9.Which of these numbers is divisible by 11? (a) 1086 (b) 1078 (c) 1072 (d) 1094
  10. 10.What is the smallest number you must add to 271 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  11. 11.What is the remainder when 143 is divided by 9? (a) 8 (b) 1 (c) 6 (d) 7
  12. 12.What is the remainder when 207 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
  13. 13.What is the units digit of 9^39? (a) 7 (b) 1 (c) 8 (d) 9
  14. 14.What is the smallest number you must add to 383 to make it divisible by 12? (a) 11 (b) 3 (c) 2 (d) 1
  15. 15.What is the smallest number you must add to 543 to make it divisible by 11? (a) 6 (b) 8 (c) 4 (d) 7
  16. 16.What is the smallest number you must add to 518 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 6
  17. 17.What is the smallest number you must add to 587 to make it divisible by 10? (a) 4 (b) 7 (c) 2 (d) 3
  18. 18.What is the units digit of 8^14? (a) 8 (b) 2 (c) 6 (d) 4
  19. 19.What is the smallest number you must add to 666 to make it divisible by 10? (a) 6 (b) 5 (c) 4 (d) 3
  20. 20.Which of these numbers is divisible by 4? (a) 2784 (b) 2783 (c) 2791 (d) 2785
  21. 21.Which of these numbers is divisible by 4? (a) 1406 (b) 1410 (c) 1401 (d) 1404
  22. 22.Which of these numbers is divisible by 4? (a) 1840 (b) 1846 (c) 1842 (d) 1837
  23. 23.Which of these numbers is divisible by 6? (a) 4405 (b) 4397 (c) 4399 (d) 4398
  24. 24.What is the units digit of 9^10? (a) 2 (b) 9 (c) 0 (d) 1
  25. 25.What is the units digit of 9^33? (a) 7 (b) 8 (c) 9 (d) 1
  26. 26.What is the remainder when 229 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
  27. 27.What is the units digit of 4^37? (a) 3 (b) 6 (c) 5 (d) 4
  28. 28.What is the smallest number you must add to 154 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  29. 29.What is the units digit of 8^46? (a) 4 (b) 6 (c) 8 (d) 2
  30. 30.What is the units digit of 8^49? (a) 2 (b) 4 (c) 8 (d) 6
  31. 31.What is the units digit of 8^53? (a) 2 (b) 8 (c) 4 (d) 6
  32. 32.What is the units digit of 2^35? (a) 2 (b) 8 (c) 6 (d) 4
  33. 33.What is the units digit of 2^43? (a) 6 (b) 4 (c) 2 (d) 8
  34. 34.What is the smallest number you must add to 223 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
  35. 35.What is the smallest number you must add to 392 to make it divisible by 6? (a) 2 (b) 4 (c) 3 (d) 5
  36. 36.Which of these numbers is divisible by 11? (a) 2099 (b) 2116 (c) 2109 (d) 2101
  37. 37.Which of these numbers is divisible by 8? (a) 2238 (b) 2222 (c) 2227 (d) 2224
  38. 38.What is the remainder when 87 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 4
  39. 39.What is the smallest number you must add to 506 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 2
  40. 40.Which of these numbers is divisible by 4? (a) 620 (b) 618 (c) 626 (d) 621

Answer key

  1. 729
  2. 4
  3. 5
  4. 2
  5. 1
  6. 2
  7. 1
  8. 1
  9. 1078
  10. 1
  11. 8
  12. 3
  13. 9
  14. 1
  15. 7
  16. 2
  17. 3
  18. 4
  19. 4
  20. 2784
  21. 1404
  22. 1840
  23. 4398
  24. 1
  25. 9
  26. 4
  27. 4
  28. 2
  29. 4
  30. 8
  31. 8
  32. 8
  33. 8
  34. 2
  35. 4
  36. 2101
  37. 2224
  38. 3
  39. 7
  40. 620

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