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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 650 · 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 349 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 2
  2. 2.What is the smallest number you must add to 61 to make it divisible by 3? (a) 2 (b) 1
  3. 3.What is the smallest number you must add to 236 to make it divisible by 3? (a) 1 (b) 2
  4. 4.Which of these numbers is divisible by 4? (a) 3479 (b) 3476 (c) 3482 (d) 3475
  5. 5.Which of these numbers is divisible by 8? (a) 2700 (b) 2689 (c) 2688 (d) 2687
  6. 6.What is the remainder when 119 is divided by 10? (a) 8 (b) 1 (c) 7 (d) 9
  7. 7.What is the smallest number you must add to 170 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
  8. 8.What is the smallest number you must add to 208 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
  9. 9.What is the units digit of 8^23? (a) 8 (b) 2 (c) 4 (d) 6
  10. 10.What is the units digit of 7^22? (a) 9 (b) 3 (c) 7 (d) 1
  11. 11.What is the remainder when 150 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
  12. 12.What is the smallest number you must add to 296 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  13. 13.What is the remainder when 392 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
  14. 14.What is the smallest number you must add to 204 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
  15. 15.What is the units digit of 8^32? (a) 4 (b) 8 (c) 6 (d) 2
  16. 16.What is the smallest number you must add to 280 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 1
  17. 17.What is the remainder when 624 is divided by 9? (a) 4 (b) 3 (c) 6 (d) 2
  18. 18.What is the units digit of 7^32? (a) 1 (b) 9 (c) 7 (d) 3
  19. 19.What is the smallest number you must add to 166 to make it divisible by 3? (a) 1 (b) 2
  20. 20.What is the remainder when 652 is divided by 11? (a) 2 (b) 4 (c) 3 (d) 8
  21. 21.What is the units digit of 7^24? (a) 1 (b) 9 (c) 3 (d) 7
  22. 22.What is the remainder when 166 is divided by 3? (a) 1 (b) 2 (c) 0
  23. 23.Which of these numbers is divisible by 8? (a) 887 (b) 889 (c) 888 (d) 903
  24. 24.What is the remainder when 225 is divided by 11? (a) 6 (b) 5 (c) 4 (d) 7
  25. 25.What is the remainder when 163 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
  26. 26.What is the units digit of 3^52? (a) 3 (b) 7 (c) 9 (d) 1
  27. 27.What is the remainder when 195 is divided by 9? (a) 3 (b) 7 (c) 5 (d) 6
  28. 28.What is the smallest number you must add to 138 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  29. 29.Which of these numbers is divisible by 9? (a) 2608 (b) 2597 (c) 2615 (d) 2601
  30. 30.What is the remainder when 293 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
  31. 31.Which of these numbers is divisible by 11? (a) 7705 (b) 7689 (c) 7690 (d) 7681
  32. 32.What is the units digit of 4^19? (a) 3 (b) 4 (c) 5 (d) 6
  33. 33.What is the units digit of 7^38? (a) 3 (b) 7 (c) 9 (d) 1
  34. 34.What is the remainder when 82 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 2
  35. 35.What is the units digit of 8^33? (a) 2 (b) 4 (c) 6 (d) 8
  36. 36.What is the remainder when 234 is divided by 11? (a) 4 (b) 3 (c) 2 (d) 8
  37. 37.What is the smallest number you must add to 258 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
  38. 38.Which of these numbers is divisible by 11? (a) 6072 (b) 6089 (c) 6066 (d) 6074
  39. 39.What is the smallest number you must add to 167 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
  40. 40.What is the smallest number you must add to 95 to make it divisible by 3? (a) 2 (b) 1

Answer key

  1. 3
  2. 2
  3. 1
  4. 3476
  5. 2688
  6. 9
  7. 4
  8. 2
  9. 2
  10. 9
  11. 2
  12. 4
  13. 2
  14. 4
  15. 6
  16. 8
  17. 3
  18. 1
  19. 2
  20. 3
  21. 1
  22. 1
  23. 888
  24. 5
  25. 3
  26. 1
  27. 6
  28. 2
  29. 2601
  30. 1
  31. 7689
  32. 4
  33. 9
  34. 5
  35. 8
  36. 3
  37. 2
  38. 6072
  39. 3
  40. 1

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