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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 109 · 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 416 to make it divisible by 12? (a) 8 (b) 3 (c) 4 (d) 5
  2. 2.What is the smallest number you must add to 608 to make it divisible by 10? (a) 1 (b) 2 (c) 3 (d) 8
  3. 3.What is the remainder when 700 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 2
  4. 4.What is the smallest number you must add to 413 to make it divisible by 8? (a) 3 (b) 4 (c) 2 (d) 5
  5. 5.What is the units digit of 4^51? (a) 6 (b) 4 (c) 3 (d) 5
  6. 6.Which of these numbers is divisible by 3? (a) 1850 (b) 1844 (c) 1846 (d) 1845
  7. 7.What is the smallest number you must add to 340 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 3
  8. 8.Which of these numbers is divisible by 9? (a) 1026 (b) 1023 (c) 1031 (d) 1041
  9. 9.What is the remainder when 197 is divided by 7? (a) 2 (b) 0 (c) 6 (d) 1
  10. 10.Which of these numbers is divisible by 6? (a) 420 (b) 423 (c) 429 (d) 419
  11. 11.What is the remainder when 405 is divided by 12? (a) 10 (b) 3 (c) 9 (d) 8
  12. 12.What is the smallest number you must add to 194 to make it divisible by 3? (a) 2 (b) 1
  13. 13.What is the smallest number you must add to 776 to make it divisible by 10? (a) 6 (b) 3 (c) 5 (d) 4
  14. 14.Which of these numbers is divisible by 11? (a) 2084 (b) 2099 (c) 2108 (d) 2090
  15. 15.What is the remainder when 136 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 0
  16. 16.What is the units digit of 2^15? (a) 2 (b) 8 (c) 6 (d) 4
  17. 17.What is the units digit of 3^32? (a) 3 (b) 7 (c) 9 (d) 1
  18. 18.What is the units digit of 4^41? (a) 6 (b) 5 (c) 4 (d) 3
  19. 19.What is the smallest number you must add to 230 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  20. 20.What is the smallest number you must add to 218 to make it divisible by 3? (a) 2 (b) 1
  21. 21.What is the remainder when 844 is divided by 11? (a) 9 (b) 8 (c) 7 (d) 3
  22. 22.What is the units digit of 3^30? (a) 9 (b) 3 (c) 7 (d) 1
  23. 23.What is the remainder when 273 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
  24. 24.Which of these numbers is divisible by 4? (a) 3206 (b) 3203 (c) 3199 (d) 3200
  25. 25.What is the smallest number you must add to 549 to make it divisible by 7? (a) 3 (b) 5 (c) 6 (d) 4
  26. 26.What is the remainder when 289 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
  27. 27.What is the units digit of 3^53? (a) 1 (b) 3 (c) 9 (d) 7
  28. 28.What is the smallest number you must add to 167 to make it divisible by 3? (a) 2 (b) 1
  29. 29.Which of these numbers is divisible by 4? (a) 1826 (b) 1821 (c) 1829 (d) 1824
  30. 30.What is the smallest number you must add to 366 to make it divisible by 8? (a) 2 (b) 1 (c) 3 (d) 6
  31. 31.What is the units digit of 8^45? (a) 2 (b) 4 (c) 8 (d) 6
  32. 32.Which of these numbers is divisible by 4? (a) 2973 (b) 2976 (c) 2983 (d) 2978
  33. 33.What is the smallest number you must add to 247 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  34. 34.What is the units digit of 4^35? (a) 4 (b) 5 (c) 3 (d) 6
  35. 35.Which of these numbers is divisible by 8? (a) 5938 (b) 5954 (c) 5944 (d) 5951
  36. 36.What is the units digit of 3^36? (a) 9 (b) 3 (c) 1 (d) 7
  37. 37.What is the smallest number you must add to 173 to make it divisible by 3? (a) 1 (b) 2
  38. 38.What is the remainder when 43 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
  39. 39.What is the units digit of 8^51? (a) 8 (b) 6 (c) 2 (d) 4
  40. 40.Which of these numbers is divisible by 9? (a) 3028 (b) 3024 (c) 3019 (d) 3034

Answer key

  1. 4
  2. 2
  3. 7
  4. 3
  5. 4
  6. 1845
  7. 3
  8. 1026
  9. 1
  10. 420
  11. 9
  12. 1
  13. 4
  14. 2090
  15. 1
  16. 8
  17. 1
  18. 4
  19. 2
  20. 1
  21. 8
  22. 9
  23. 3
  24. 3200
  25. 4
  26. 1
  27. 3
  28. 1
  29. 1824
  30. 2
  31. 8
  32. 2976
  33. 1
  34. 4
  35. 5944
  36. 1
  37. 1
  38. 3
  39. 2
  40. 3024

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