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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 22 · 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 140 to make it divisible by 9? (a) 3 (b) 5 (c) 6 (d) 4
  2. 2.Which of these numbers is divisible by 8? (a) 936 (b) 941 (c) 945 (d) 929
  3. 3.What is the remainder when 38 is divided by 3? (a) 0 (b) 1 (c) 2
  4. 4.What is the smallest number you must add to 415 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 2
  5. 5.Which of these numbers is divisible by 6? (a) 1560 (b) 1567 (c) 1559 (d) 1563
  6. 6.What is the smallest number you must add to 734 to make it divisible by 12? (a) 11 (b) 10 (c) 9 (d) 2
  7. 7.Which of these numbers is divisible by 8? (a) 6619 (b) 6636 (c) 6631 (d) 6624
  8. 8.What is the units digit of 3^53? (a) 9 (b) 3 (c) 1 (d) 7
  9. 9.Which of these numbers is divisible by 8? (a) 4724 (b) 4732 (c) 4719 (d) 4720
  10. 10.What is the remainder when 287 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
  11. 11.What is the smallest number you must add to 674 to make it divisible by 12? (a) 11 (b) 10 (c) 2 (d) 9
  12. 12.What is the smallest number you must add to 158 to make it divisible by 3? (a) 2 (b) 1
  13. 13.What is the units digit of 3^23? (a) 3 (b) 7 (c) 1 (d) 9
  14. 14.Which of these numbers is divisible by 11? (a) 2269 (b) 2245 (c) 2255 (d) 2261
  15. 15.Which of these numbers is divisible by 9? (a) 3607 (b) 3587 (c) 3591 (d) 3594
  16. 16.What is the units digit of 3^32? (a) 3 (b) 9 (c) 7 (d) 1
  17. 17.Which of these numbers is divisible by 4? (a) 225 (b) 233 (c) 231 (d) 228
  18. 18.What is the smallest number you must add to 211 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
  19. 19.What is the remainder when 224 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
  20. 20.What is the remainder when 149 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
  21. 21.What is the remainder when 115 is divided by 8? (a) 5 (b) 3 (c) 2 (d) 4
  22. 22.What is the units digit of 2^13? (a) 4 (b) 8 (c) 6 (d) 2
  23. 23.Which of these numbers is divisible by 4? (a) 2315 (b) 2322 (c) 2319 (d) 2316
  24. 24.What is the smallest number you must add to 78 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
  25. 25.What is the remainder when 242 is divided by 8? (a) 6 (b) 3 (c) 2 (d) 1
  26. 26.Which of these numbers is divisible by 8? (a) 2569 (b) 2580 (c) 2568 (d) 2566
  27. 27.Which of these numbers is divisible by 6? (a) 5106 (b) 5107 (c) 5102 (d) 5116
  28. 28.What is the remainder when 149 is divided by 7? (a) 3 (b) 1 (c) 2 (d) 5
  29. 29.What is the units digit of 4^58? (a) 7 (b) 6 (c) 5 (d) 4
  30. 30.What is the units digit of 3^11? (a) 9 (b) 7 (c) 1 (d) 3
  31. 31.What is the smallest number you must add to 588 to make it divisible by 10? (a) 3 (b) 2 (c) 8 (d) 1
  32. 32.What is the smallest number you must add to 434 to make it divisible by 11? (a) 8 (b) 6 (c) 5 (d) 7
  33. 33.What is the smallest number you must add to 259 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
  34. 34.What is the remainder when 241 is divided by 3? (a) 1 (b) 2 (c) 0
  35. 35.What is the smallest number you must add to 396 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
  36. 36.What is the remainder when 290 is divided by 12? (a) 3 (b) 2 (c) 10 (d) 1
  37. 37.Which of these numbers is divisible by 3? (a) 2403 (b) 2402 (c) 2407 (d) 2405
  38. 38.Which of these numbers is divisible by 3? (a) 566 (b) 569 (c) 572 (d) 567
  39. 39.What is the smallest number you must add to 421 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
  40. 40.Which of these numbers is divisible by 6? (a) 3125 (b) 3120 (c) 3131 (d) 3117

Answer key

  1. 4
  2. 936
  3. 2
  4. 5
  5. 1560
  6. 10
  7. 6624
  8. 3
  9. 4720
  10. 2
  11. 10
  12. 1
  13. 7
  14. 2255
  15. 3591
  16. 1
  17. 228
  18. 4
  19. 4
  20. 4
  21. 3
  22. 2
  23. 2316
  24. 2
  25. 2
  26. 2568
  27. 5106
  28. 2
  29. 6
  30. 7
  31. 2
  32. 6
  33. 1
  34. 1
  35. 4
  36. 2
  37. 2403
  38. 567
  39. 2
  40. 3120

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