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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 571 · 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 smallest number you must add to 119 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  2. 2.Which of these numbers is divisible by 11? (a) 9748 (b) 9743 (c) 9735 (d) 9726
  3. 3.What is the smallest number you must add to 795 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
  4. 4.What is the smallest number you must add to 216 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
  5. 5.Which of these numbers is divisible by 6? (a) 5394 (b) 5405 (c) 5391 (d) 5396
  6. 6.What is the remainder when 64 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
  7. 7.What is the remainder when 263 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
  8. 8.Which of these numbers is divisible by 3? (a) 1965 (b) 1966 (c) 1969 (d) 1963
  9. 9.What is the smallest number you must add to 730 to make it divisible by 11? (a) 4 (b) 8 (c) 6 (d) 7
  10. 10.What is the smallest number you must add to 314 to make it divisible by 9? (a) 1 (b) 8 (c) 3 (d) 2
  11. 11.Which of these numbers is divisible by 6? (a) 4658 (b) 4648 (c) 4650 (d) 4653
  12. 12.What is the units digit of 9^17? (a) 1 (b) 9 (c) 7 (d) 8
  13. 13.What is the remainder when 233 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
  14. 14.Which of these numbers is divisible by 4? (a) 1217 (b) 1223 (c) 1216 (d) 1215
  15. 15.Which of these numbers is divisible by 9? (a) 2701 (b) 2690 (c) 2691 (d) 2694
  16. 16.What is the remainder when 618 is divided by 11? (a) 2 (b) 1 (c) 9 (d) 3
  17. 17.What is the units digit of 8^53? (a) 6 (b) 2 (c) 8 (d) 4
  18. 18.What is the units digit of 3^32? (a) 3 (b) 1 (c) 9 (d) 7
  19. 19.Which of these numbers is divisible by 8? (a) 5713 (b) 5720 (c) 5733 (d) 5724
  20. 20.What is the smallest number you must add to 167 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  21. 21.Which of these numbers is divisible by 9? (a) 4331 (b) 4338 (c) 4340 (d) 4352
  22. 22.What is the smallest number you must add to 82 to make it divisible by 8? (a) 7 (b) 6 (c) 2 (d) 5
  23. 23.What is the smallest number you must add to 376 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  24. 24.Which of these numbers is divisible by 9? (a) 4824 (b) 4827 (c) 4823 (d) 4835
  25. 25.What is the units digit of 7^11? (a) 1 (b) 9 (c) 3 (d) 7
  26. 26.What is the smallest number you must add to 382 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 2
  27. 27.Which of these numbers is divisible by 3? (a) 1846 (b) 1843 (c) 1842 (d) 1840
  28. 28.What is the smallest number you must add to 401 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
  29. 29.Which of these numbers is divisible by 9? (a) 6781 (b) 6777 (c) 6791 (d) 6769
  30. 30.What is the smallest number you must add to 107 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  31. 31.Which of these numbers is divisible by 6? (a) 4788 (b) 4789 (c) 4796 (d) 4783
  32. 32.Which of these numbers is divisible by 4? (a) 1634 (b) 1631 (c) 1626 (d) 1628
  33. 33.Which of these numbers is divisible by 4? (a) 298 (b) 301 (c) 296 (d) 295
  34. 34.Which of these numbers is divisible by 9? (a) 7671 (b) 7668 (c) 7678 (d) 7660
  35. 35.What is the smallest number you must add to 57 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  36. 36.Which of these numbers is divisible by 9? (a) 2575 (b) 2585 (c) 2567 (d) 2574
  37. 37.What is the smallest number you must add to 70 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  38. 38.What is the units digit of 2^21? (a) 6 (b) 2 (c) 8 (d) 4
  39. 39.What is the smallest number you must add to 878 to make it divisible by 12? (a) 9 (b) 10 (c) 11 (d) 2
  40. 40.What is the remainder when 45 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1

Answer key

  1. 1
  2. 9735
  3. 5
  4. 4
  5. 5394
  6. 4
  7. 3
  8. 1965
  9. 7
  10. 1
  11. 4650
  12. 9
  13. 3
  14. 1216
  15. 2691
  16. 2
  17. 8
  18. 1
  19. 5720
  20. 1
  21. 4338
  22. 6
  23. 4
  24. 4824
  25. 3
  26. 2
  27. 1842
  28. 4
  29. 6777
  30. 3
  31. 4788
  32. 1628
  33. 296
  34. 7668
  35. 3
  36. 2574
  37. 2
  38. 2
  39. 10
  40. 1

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