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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 688 · 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.Which of these numbers is divisible by 3? (a) 1196 (b) 1191 (c) 1189 (d) 1192
  2. 2.What is the remainder when 436 is divided by 6? (a) 2 (b) 5 (c) 3 (d) 4
  3. 3.What is the units digit of 8^13? (a) 2 (b) 6 (c) 4 (d) 8
  4. 4.What is the smallest number you must add to 316 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
  5. 5.What is the remainder when 428 is divided by 6? (a) 2 (b) 1 (c) 4 (d) 3
  6. 6.What is the units digit of 3^19? (a) 9 (b) 7 (c) 3 (d) 1
  7. 7.Which of these numbers is divisible by 4? (a) 2980 (b) 2978 (c) 2983 (d) 2986
  8. 8.What is the remainder when 519 is divided by 8? (a) 6 (b) 1 (c) 7 (d) 5
  9. 9.What is the remainder when 218 is divided by 3? (a) 1 (b) 0 (c) 2
  10. 10.What is the remainder when 171 is divided by 11? (a) 5 (b) 8 (c) 6 (d) 7
  11. 11.What is the units digit of 8^35? (a) 4 (b) 6 (c) 8 (d) 2
  12. 12.What is the remainder when 645 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
  13. 13.What is the units digit of 3^22? (a) 1 (b) 7 (c) 9 (d) 3
  14. 14.What is the smallest number you must add to 231 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
  15. 15.What is the remainder when 646 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 7
  16. 16.What is the remainder when 177 is divided by 12? (a) 9 (b) 8 (c) 10 (d) 3
  17. 17.What is the smallest number you must add to 613 to make it divisible by 10? (a) 3 (b) 6 (c) 8 (d) 7
  18. 18.What is the remainder when 122 is divided by 9? (a) 7 (b) 4 (c) 5 (d) 6
  19. 19.What is the smallest number you must add to 197 to make it divisible by 3? (a) 2 (b) 1
  20. 20.What is the units digit of 4^37? (a) 3 (b) 4 (c) 6 (d) 5
  21. 21.What is the smallest number you must add to 140 to make it divisible by 3? (a) 1 (b) 2
  22. 22.What is the units digit of 3^11? (a) 3 (b) 1 (c) 9 (d) 7
  23. 23.What is the smallest number you must add to 798 to make it divisible by 10? (a) 8 (b) 2 (c) 1 (d) 3
  24. 24.Which of these numbers is divisible by 3? (a) 2160 (b) 2164 (c) 2158 (d) 2162
  25. 25.What is the smallest number you must add to 319 to make it divisible by 6? (a) 5 (b) 1 (c) 4 (d) 3
  26. 26.What is the units digit of 7^15? (a) 9 (b) 1 (c) 3 (d) 7
  27. 27.What is the units digit of 2^50? (a) 8 (b) 4 (c) 6 (d) 2
  28. 28.What is the units digit of 9^35? (a) 8 (b) 7 (c) 9 (d) 1
  29. 29.What is the units digit of 3^44? (a) 7 (b) 3 (c) 1 (d) 9
  30. 30.What is the remainder when 168 is divided by 9? (a) 6 (b) 3 (c) 5 (d) 7
  31. 31.What is the smallest number you must add to 147 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  32. 32.What is the smallest number you must add to 113 to make it divisible by 3? (a) 2 (b) 1
  33. 33.What is the smallest number you must add to 43 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  34. 34.What is the units digit of 4^13? (a) 5 (b) 4 (c) 3 (d) 6
  35. 35.Which of these numbers is divisible by 8? (a) 1184 (b) 1193 (c) 1187 (d) 1183
  36. 36.What is the remainder when 91 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
  37. 37.What is the remainder when 935 is divided by 12? (a) 11 (b) 9 (c) 10 (d) 1
  38. 38.What is the remainder when 116 is divided by 10? (a) 7 (b) 5 (c) 6 (d) 4
  39. 39.What is the units digit of 4^21? (a) 3 (b) 5 (c) 6 (d) 4
  40. 40.What is the smallest number you must add to 802 to make it divisible by 10? (a) 2 (b) 7 (c) 8 (d) 9

Answer key

  1. 1191
  2. 4
  3. 8
  4. 4
  5. 2
  6. 7
  7. 2980
  8. 7
  9. 2
  10. 6
  11. 2
  12. 5
  13. 9
  14. 4
  15. 6
  16. 9
  17. 7
  18. 5
  19. 1
  20. 4
  21. 1
  22. 7
  23. 2
  24. 2160
  25. 5
  26. 3
  27. 4
  28. 9
  29. 1
  30. 6
  31. 1
  32. 1
  33. 1
  34. 4
  35. 1184
  36. 3
  37. 11
  38. 6
  39. 4
  40. 8

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