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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 264 · 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 units digit of 7^30? (a) 3 (b) 7 (c) 9 (d) 1
  2. 2.What is the remainder when 676 is divided by 11? (a) 6 (b) 4 (c) 7 (d) 5
  3. 3.Which of these numbers is divisible by 8? (a) 6858 (b) 6864 (c) 6874 (d) 6868
  4. 4.What is the smallest number you must add to 173 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  5. 5.Which of these numbers is divisible by 9? (a) 2186 (b) 2161 (c) 2169 (d) 2173
  6. 6.What is the units digit of 9^50? (a) 0 (b) 2 (c) 1 (d) 9
  7. 7.What is the remainder when 529 is divided by 12? (a) 11 (b) 2 (c) 0 (d) 1
  8. 8.What is the smallest number you must add to 53 to make it divisible by 3? (a) 2 (b) 1
  9. 9.What is the smallest number you must add to 194 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
  10. 10.What is the remainder when 202 is divided by 3? (a) 1 (b) 0 (c) 2
  11. 11.What is the remainder when 820 is divided by 12? (a) 8 (b) 4 (c) 3 (d) 5
  12. 12.Which of these numbers is divisible by 9? (a) 5685 (b) 5688 (c) 5694 (d) 5700
  13. 13.Which of these numbers is divisible by 9? (a) 7965 (b) 7967 (c) 7962 (d) 7976
  14. 14.Which of these numbers is divisible by 6? (a) 1673 (b) 1667 (c) 1679 (d) 1668
  15. 15.What is the units digit of 8^43? (a) 6 (b) 4 (c) 2 (d) 8
  16. 16.What is the units digit of 7^28? (a) 9 (b) 7 (c) 3 (d) 1
  17. 17.What is the smallest number you must add to 148 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
  18. 18.What is the units digit of 7^58? (a) 3 (b) 7 (c) 1 (d) 9
  19. 19.What is the remainder when 91 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 0
  20. 20.What is the remainder when 175 is divided by 3? (a) 2 (b) 1 (c) 0
  21. 21.What is the units digit of 3^48? (a) 9 (b) 1 (c) 7 (d) 3
  22. 22.What is the smallest number you must add to 417 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 2
  23. 23.What is the units digit of 4^47? (a) 6 (b) 4 (c) 3 (d) 5
  24. 24.What is the units digit of 8^13? (a) 8 (b) 6 (c) 2 (d) 4
  25. 25.What is the units digit of 8^21? (a) 4 (b) 8 (c) 6 (d) 2
  26. 26.What is the remainder when 795 is divided by 12? (a) 2 (b) 9 (c) 4 (d) 3
  27. 27.What is the smallest number you must add to 247 to make it divisible by 9? (a) 6 (b) 5 (c) 7 (d) 4
  28. 28.What is the smallest number you must add to 620 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
  29. 29.What is the units digit of 3^10? (a) 7 (b) 1 (c) 3 (d) 9
  30. 30.Which of these numbers is divisible by 6? (a) 4278 (b) 4274 (c) 4280 (d) 4285
  31. 31.What is the smallest number you must add to 699 to make it divisible by 9? (a) 6 (b) 2 (c) 4 (d) 3
  32. 32.What is the units digit of 8^16? (a) 2 (b) 6 (c) 8 (d) 4
  33. 33.What is the smallest number you must add to 178 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
  34. 34.What is the units digit of 7^14? (a) 3 (b) 9 (c) 1 (d) 7
  35. 35.What is the units digit of 9^13? (a) 1 (b) 9 (c) 8 (d) 7
  36. 36.What is the remainder when 293 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
  37. 37.What is the smallest number you must add to 230 to make it divisible by 12? (a) 2 (b) 10 (c) 11 (d) 9
  38. 38.What is the smallest number you must add to 124 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
  39. 39.What is the smallest number you must add to 163 to make it divisible by 6? (a) 1 (b) 3 (c) 4 (d) 5
  40. 40.What is the remainder when 536 is divided by 9? (a) 5 (b) 4 (c) 6 (d) 7

Answer key

  1. 9
  2. 5
  3. 6864
  4. 3
  5. 2169
  6. 1
  7. 1
  8. 1
  9. 4
  10. 1
  11. 4
  12. 5688
  13. 7965
  14. 1668
  15. 2
  16. 1
  17. 2
  18. 9
  19. 1
  20. 1
  21. 1
  22. 3
  23. 4
  24. 8
  25. 8
  26. 3
  27. 5
  28. 4
  29. 9
  30. 4278
  31. 3
  32. 6
  33. 2
  34. 9
  35. 9
  36. 5
  37. 10
  38. 6
  39. 5
  40. 5

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