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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 746 · 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 101 to make it divisible by 7? (a) 5 (b) 3 (c) 4 (d) 6
  2. 2.Which of these numbers is divisible by 8? (a) 7001 (b) 7010 (c) 7022 (d) 7008
  3. 3.What is the units digit of 7^47? (a) 7 (b) 1 (c) 9 (d) 3
  4. 4.What is the remainder when 142 is divided by 3? (a) 2 (b) 1 (c) 0
  5. 5.What is the smallest number you must add to 187 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 4
  6. 6.What is the smallest number you must add to 410 to make it divisible by 11? (a) 8 (b) 7 (c) 3 (d) 9
  7. 7.What is the remainder when 252 is divided by 8? (a) 3 (b) 4 (c) 6 (d) 5
  8. 8.What is the units digit of 8^42? (a) 6 (b) 8 (c) 4 (d) 2
  9. 9.What is the remainder when 303 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 1
  10. 10.Which of these numbers is divisible by 11? (a) 8771 (b) 8784 (c) 8795 (d) 8778
  11. 11.Which of these numbers is divisible by 8? (a) 2762 (b) 2756 (c) 2752 (d) 2745
  12. 12.What is the smallest number you must add to 358 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  13. 13.What is the remainder when 220 is divided by 7? (a) 4 (b) 5 (c) 3 (d) 2
  14. 14.What is the smallest number you must add to 501 to make it divisible by 11? (a) 4 (b) 5 (c) 6 (d) 7
  15. 15.Which of these numbers is divisible by 4? (a) 3145 (b) 3142 (c) 3149 (d) 3144
  16. 16.What is the smallest number you must add to 394 to make it divisible by 10? (a) 5 (b) 7 (c) 6 (d) 4
  17. 17.What is the units digit of 8^23? (a) 6 (b) 4 (c) 2 (d) 8
  18. 18.Which of these numbers is divisible by 9? (a) 5975 (b) 5962 (c) 5979 (d) 5967
  19. 19.Which of these numbers is divisible by 4? (a) 2121 (b) 2125 (c) 2131 (d) 2124
  20. 20.What is the remainder when 62 is divided by 3? (a) 1 (b) 2 (c) 0
  21. 21.What is the smallest number you must add to 194 to make it divisible by 3? (a) 2 (b) 1
  22. 22.What is the units digit of 4^58? (a) 7 (b) 4 (c) 5 (d) 6
  23. 23.What is the smallest number you must add to 106 to make it divisible by 3? (a) 2 (b) 1
  24. 24.What is the remainder when 602 is divided by 8? (a) 1 (b) 3 (c) 6 (d) 2
  25. 25.What is the remainder when 117 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
  26. 26.Which of these numbers is divisible by 8? (a) 5324 (b) 5331 (c) 5339 (d) 5328
  27. 27.What is the smallest number you must add to 194 to make it divisible by 7? (a) 2 (b) 5 (c) 1 (d) 3
  28. 28.Which of these numbers is divisible by 9? (a) 1713 (b) 1708 (c) 1710 (d) 1723
  29. 29.What is the units digit of 7^12? (a) 3 (b) 9 (c) 7 (d) 1
  30. 30.What is the smallest number you must add to 177 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  31. 31.What is the smallest number you must add to 217 to make it divisible by 3? (a) 1 (b) 2
  32. 32.Which of these numbers is divisible by 3? (a) 2695 (b) 2701 (c) 2697 (d) 2699
  33. 33.What is the remainder when 454 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
  34. 34.What is the smallest number you must add to 45 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  35. 35.What is the remainder when 128 is divided by 3? (a) 1 (b) 0 (c) 2
  36. 36.Which of these numbers is divisible by 4? (a) 2530 (b) 2532 (c) 2533 (d) 2537
  37. 37.What is the remainder when 69 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
  38. 38.What is the smallest number you must add to 369 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 5
  39. 39.What is the units digit of 2^33? (a) 6 (b) 4 (c) 8 (d) 2
  40. 40.Which of these numbers is divisible by 11? (a) 6419 (b) 6434 (c) 6413 (d) 6410

Answer key

  1. 4
  2. 7008
  3. 3
  4. 1
  5. 5
  6. 8
  7. 4
  8. 4
  9. 2
  10. 8778
  11. 2752
  12. 2
  13. 3
  14. 5
  15. 3144
  16. 6
  17. 2
  18. 5967
  19. 2124
  20. 2
  21. 1
  22. 6
  23. 2
  24. 2
  25. 1
  26. 5328
  27. 2
  28. 1710
  29. 1
  30. 3
  31. 2
  32. 2697
  33. 4
  34. 3
  35. 2
  36. 2532
  37. 3
  38. 3
  39. 2
  40. 6413

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