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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 301 · 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 704 to make it divisible by 10? (a) 6 (b) 4 (c) 7 (d) 5
  2. 2.What is the smallest number you must add to 254 to make it divisible by 11? (a) 1 (b) 10 (c) 9 (d) 8
  3. 3.What is the smallest number you must add to 322 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
  4. 4.Which of these numbers is divisible by 8? (a) 1592 (b) 1585 (c) 1597 (d) 1606
  5. 5.What is the units digit of 3^41? (a) 1 (b) 3 (c) 9 (d) 7
  6. 6.Which of these numbers is divisible by 4? (a) 2575 (b) 2577 (c) 2576 (d) 2581
  7. 7.What is the units digit of 4^23? (a) 3 (b) 6 (c) 4 (d) 5
  8. 8.Which of these numbers is divisible by 8? (a) 2175 (b) 2178 (c) 2168 (d) 2163
  9. 9.Which of these numbers is divisible by 8? (a) 1184 (b) 1196 (c) 1182 (d) 1191
  10. 10.What is the smallest number you must add to 285 to make it divisible by 7? (a) 1 (b) 2 (c) 5 (d) 3
  11. 11.Which of these numbers is divisible by 4? (a) 1818 (b) 1814 (c) 1809 (d) 1812
  12. 12.What is the remainder when 133 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
  13. 13.What is the smallest number you must add to 739 to make it divisible by 11? (a) 2 (b) 10 (c) 8 (d) 9
  14. 14.What is the smallest number you must add to 101 to make it divisible by 9? (a) 2 (b) 8 (c) 6 (d) 7
  15. 15.What is the smallest number you must add to 511 to make it divisible by 8? (a) 7 (b) 3 (c) 1 (d) 2
  16. 16.What is the remainder when 884 is divided by 12? (a) 4 (b) 7 (c) 8 (d) 9
  17. 17.What is the remainder when 174 is divided by 10? (a) 3 (b) 5 (c) 4 (d) 6
  18. 18.What is the units digit of 9^36? (a) 9 (b) 1 (c) 0 (d) 2
  19. 19.What is the remainder when 260 is divided by 12? (a) 9 (b) 8 (c) 4 (d) 7
  20. 20.What is the remainder when 362 is divided by 8? (a) 2 (b) 1 (c) 3 (d) 6
  21. 21.What is the remainder when 460 is divided by 7? (a) 2 (b) 4 (c) 5 (d) 6
  22. 22.Which of these numbers is divisible by 6? (a) 3059 (b) 3061 (c) 3071 (d) 3060
  23. 23.What is the units digit of 7^41? (a) 9 (b) 1 (c) 7 (d) 3
  24. 24.What is the remainder when 326 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 1
  25. 25.What is the remainder when 42 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
  26. 26.Which of these numbers is divisible by 6? (a) 1227 (b) 1218 (c) 1215 (d) 1220
  27. 27.What is the remainder when 686 is divided by 11? (a) 7 (b) 4 (c) 3 (d) 5
  28. 28.Which of these numbers is divisible by 6? (a) 3979 (b) 3978 (c) 3974 (d) 3985
  29. 29.What is the smallest number you must add to 136 to make it divisible by 3? (a) 2 (b) 1
  30. 30.What is the remainder when 167 is divided by 10? (a) 7 (b) 3 (c) 8 (d) 6
  31. 31.Which of these numbers is divisible by 8? (a) 2620 (b) 2633 (c) 2624 (d) 2625
  32. 32.What is the smallest number you must add to 82 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  33. 33.What is the units digit of 7^34? (a) 9 (b) 7 (c) 1 (d) 3
  34. 34.What is the remainder when 583 is divided by 12? (a) 8 (b) 5 (c) 6 (d) 7
  35. 35.What is the smallest number you must add to 402 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  36. 36.What is the units digit of 7^10? (a) 7 (b) 3 (c) 9 (d) 1
  37. 37.What is the smallest number you must add to 217 to make it divisible by 3? (a) 1 (b) 2
  38. 38.Which of these numbers is divisible by 4? (a) 3526 (b) 3518 (c) 3523 (d) 3520
  39. 39.Which of these numbers is divisible by 11? (a) 5416 (b) 5433 (c) 5423 (d) 5435
  40. 40.What is the smallest number you must add to 203 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1

Answer key

  1. 6
  2. 10
  3. 3
  4. 1592
  5. 3
  6. 2576
  7. 4
  8. 2168
  9. 1184
  10. 2
  11. 1812
  12. 1
  13. 9
  14. 7
  15. 1
  16. 8
  17. 4
  18. 1
  19. 8
  20. 2
  21. 5
  22. 3060
  23. 7
  24. 2
  25. 2
  26. 1218
  27. 4
  28. 3978
  29. 2
  30. 7
  31. 2624
  32. 2
  33. 9
  34. 7
  35. 3
  36. 9
  37. 2
  38. 3520
  39. 5423
  40. 2

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