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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 651 · 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 8^20? (a) 4 (b) 6 (c) 2 (d) 8
  2. 2.What is the remainder when 101 is divided by 8? (a) 6 (b) 5 (c) 4 (d) 3
  3. 3.Which of these numbers is divisible by 8? (a) 2791 (b) 2792 (c) 2806 (d) 2793
  4. 4.What is the remainder when 386 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
  5. 5.What is the remainder when 227 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
  6. 6.Which of these numbers is divisible by 8? (a) 830 (b) 838 (c) 841 (d) 832
  7. 7.What is the remainder when 327 is divided by 7? (a) 2 (b) 5 (c) 6 (d) 4
  8. 8.What is the units digit of 4^13? (a) 5 (b) 3 (c) 4 (d) 6
  9. 9.What is the units digit of 4^47? (a) 4 (b) 5 (c) 3 (d) 6
  10. 10.What is the smallest number you must add to 558 to make it divisible by 10? (a) 1 (b) 3 (c) 8 (d) 2
  11. 11.What is the remainder when 191 is divided by 8? (a) 5 (b) 1 (c) 7 (d) 6
  12. 12.Which of these numbers is divisible by 4? (a) 1223 (b) 1215 (c) 1216 (d) 1218
  13. 13.What is the remainder when 91 is divided by 3? (a) 0 (b) 1 (c) 2
  14. 14.Which of these numbers is divisible by 4? (a) 1067 (b) 1061 (c) 1057 (d) 1060
  15. 15.Which of these numbers is divisible by 3? (a) 966 (b) 965 (c) 967 (d) 971
  16. 16.What is the remainder when 356 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
  17. 17.What is the units digit of 9^40? (a) 2 (b) 1 (c) 0 (d) 9
  18. 18.What is the smallest number you must add to 731 to make it divisible by 10? (a) 8 (b) 9 (c) 1 (d) 7
  19. 19.What is the remainder when 141 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
  20. 20.What is the units digit of 2^30? (a) 8 (b) 2 (c) 4 (d) 6
  21. 21.What is the smallest number you must add to 214 to make it divisible by 8? (a) 2 (b) 3 (c) 6 (d) 1
  22. 22.What is the units digit of 3^50? (a) 3 (b) 7 (c) 1 (d) 9
  23. 23.Which of these numbers is divisible by 11? (a) 1359 (b) 1334 (c) 1342 (d) 1344
  24. 24.What is the smallest number you must add to 98 to make it divisible by 3? (a) 2 (b) 1
  25. 25.What is the remainder when 167 is divided by 8? (a) 6 (b) 7 (c) 1 (d) 5
  26. 26.What is the smallest number you must add to 400 to make it divisible by 9? (a) 5 (b) 7 (c) 4 (d) 6
  27. 27.What is the units digit of 9^48? (a) 9 (b) 0 (c) 1 (d) 2
  28. 28.What is the remainder when 196 is divided by 8? (a) 6 (b) 3 (c) 4 (d) 5
  29. 29.What is the units digit of 2^22? (a) 4 (b) 6 (c) 8 (d) 2
  30. 30.What is the smallest number you must add to 75 to make it divisible by 7? (a) 5 (b) 1 (c) 3 (d) 2
  31. 31.What is the units digit of 2^16? (a) 2 (b) 8 (c) 6 (d) 4
  32. 32.What is the smallest number you must add to 391 to make it divisible by 9? (a) 4 (b) 6 (c) 7 (d) 5
  33. 33.What is the smallest number you must add to 674 to make it divisible by 11? (a) 7 (b) 9 (c) 3 (d) 8
  34. 34.What is the remainder when 203 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 2
  35. 35.What is the units digit of 3^32? (a) 7 (b) 9 (c) 3 (d) 1
  36. 36.What is the units digit of 3^51? (a) 1 (b) 7 (c) 9 (d) 3
  37. 37.What is the smallest number you must add to 218 to make it divisible by 3? (a) 2 (b) 1
  38. 38.What is the units digit of 2^56? (a) 6 (b) 4 (c) 2 (d) 8
  39. 39.Which of these numbers is divisible by 4? (a) 2388 (b) 2386 (c) 2394 (d) 2390
  40. 40.What is the remainder when 139 is divided by 3? (a) 0 (b) 1 (c) 2

Answer key

  1. 6
  2. 5
  3. 2792
  4. 6
  5. 2
  6. 832
  7. 5
  8. 4
  9. 4
  10. 2
  11. 7
  12. 1216
  13. 1
  14. 1060
  15. 966
  16. 1
  17. 1
  18. 9
  19. 1
  20. 4
  21. 2
  22. 9
  23. 1342
  24. 1
  25. 7
  26. 5
  27. 1
  28. 4
  29. 4
  30. 2
  31. 6
  32. 5
  33. 8
  34. 3
  35. 1
  36. 7
  37. 1
  38. 6
  39. 2388
  40. 1

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