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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 896 · 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 213 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  2. 2.What is the smallest number you must add to 114 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  3. 3.What is the units digit of 2^59? (a) 2 (b) 8 (c) 6 (d) 4
  4. 4.What is the remainder when 34 is divided by 3? (a) 0 (b) 2 (c) 1
  5. 5.What is the units digit of 7^30? (a) 7 (b) 9 (c) 1 (d) 3
  6. 6.What is the smallest number you must add to 207 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
  7. 7.Which of these numbers is divisible by 9? (a) 4786 (b) 4774 (c) 4779 (d) 4796
  8. 8.Which of these numbers is divisible by 11? (a) 6687 (b) 6666 (c) 6661 (d) 6672
  9. 9.What is the smallest number you must add to 297 to make it divisible by 12? (a) 2 (b) 4 (c) 9 (d) 3
  10. 10.Which of these numbers is divisible by 3? (a) 188 (b) 189 (c) 191 (d) 194
  11. 11.What is the remainder when 507 is divided by 10? (a) 8 (b) 7 (c) 6 (d) 3
  12. 12.What is the units digit of 4^11? (a) 3 (b) 4 (c) 6 (d) 5
  13. 13.What is the remainder when 687 is divided by 11? (a) 5 (b) 6 (c) 7 (d) 4
  14. 14.What is the units digit of 8^20? (a) 8 (b) 2 (c) 4 (d) 6
  15. 15.What is the smallest number you must add to 208 to make it divisible by 3? (a) 1 (b) 2
  16. 16.Which of these numbers is divisible by 6? (a) 4498 (b) 4493 (c) 4487 (d) 4488
  17. 17.What is the smallest number you must add to 368 to make it divisible by 11? (a) 6 (b) 8 (c) 7 (d) 5
  18. 18.What is the smallest number you must add to 34 to make it divisible by 3? (a) 1 (b) 2
  19. 19.Which of these numbers is divisible by 4? (a) 2386 (b) 2381 (c) 2379 (d) 2380
  20. 20.What is the smallest number you must add to 110 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  21. 21.Which of these numbers is divisible by 3? (a) 842 (b) 835 (c) 839 (d) 837
  22. 22.Which of these numbers is divisible by 9? (a) 1998 (b) 1999 (c) 1994 (d) 2009
  23. 23.What is the smallest number you must add to 114 to make it divisible by 9? (a) 2 (b) 4 (c) 3 (d) 6
  24. 24.What is the smallest number you must add to 206 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
  25. 25.Which of these numbers is divisible by 9? (a) 996 (b) 1009 (c) 1006 (d) 999
  26. 26.What is the units digit of 4^45? (a) 6 (b) 3 (c) 4 (d) 5
  27. 27.What is the remainder when 156 is divided by 11? (a) 2 (b) 3 (c) 1 (d) 9
  28. 28.What is the smallest number you must add to 80 to make it divisible by 3? (a) 2 (b) 1
  29. 29.Which of these numbers is divisible by 3? (a) 161 (b) 166 (c) 164 (d) 162
  30. 30.What is the smallest number you must add to 607 to make it divisible by 8? (a) 1 (b) 3 (c) 7 (d) 2
  31. 31.What is the smallest number you must add to 103 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 5
  32. 32.Which of these numbers is divisible by 9? (a) 1836 (b) 1833 (c) 1839 (d) 1853
  33. 33.What is the units digit of 7^47? (a) 1 (b) 9 (c) 7 (d) 3
  34. 34.What is the smallest number you must add to 71 to make it divisible by 3? (a) 1 (b) 2
  35. 35.What is the smallest number you must add to 470 to make it divisible by 12? (a) 11 (b) 9 (c) 10 (d) 2
  36. 36.What is the smallest number you must add to 356 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
  37. 37.What is the units digit of 4^37? (a) 3 (b) 5 (c) 4 (d) 6
  38. 38.What is the remainder when 694 is divided by 10? (a) 3 (b) 6 (c) 4 (d) 5
  39. 39.What is the units digit of 9^56? (a) 9 (b) 2 (c) 1 (d) 0
  40. 40.Which of these numbers is divisible by 3? (a) 2464 (b) 2467 (c) 2463 (d) 2462

Answer key

  1. 3
  2. 2
  3. 8
  4. 1
  5. 9
  6. 3
  7. 4779
  8. 6666
  9. 3
  10. 189
  11. 7
  12. 4
  13. 5
  14. 6
  15. 2
  16. 4488
  17. 6
  18. 2
  19. 2380
  20. 2
  21. 837
  22. 1998
  23. 3
  24. 4
  25. 999
  26. 4
  27. 2
  28. 1
  29. 162
  30. 1
  31. 5
  32. 1836
  33. 3
  34. 1
  35. 10
  36. 4
  37. 4
  38. 4
  39. 1
  40. 2463

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