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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 624 · 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 181 to make it divisible by 8? (a) 2 (b) 4 (c) 3 (d) 5
  2. 2.What is the units digit of 2^54? (a) 4 (b) 6 (c) 8 (d) 2
  3. 3.What is the smallest number you must add to 673 to make it divisible by 10? (a) 7 (b) 3 (c) 8 (d) 6
  4. 4.What is the smallest number you must add to 42 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  5. 5.What is the smallest number you must add to 203 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
  6. 6.What is the smallest number you must add to 273 to make it divisible by 4? (a) 1 (b) 2 (c) 3
  7. 7.Which of these numbers is divisible by 9? (a) 2385 (b) 2401 (c) 2387 (d) 2379
  8. 8.What is the remainder when 106 is divided by 9? (a) 7 (b) 8 (c) 6 (d) 2
  9. 9.What is the smallest number you must add to 451 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 1
  10. 10.What is the remainder when 471 is divided by 7? (a) 5 (b) 3 (c) 1 (d) 2
  11. 11.What is the units digit of 8^46? (a) 2 (b) 6 (c) 4 (d) 8
  12. 12.What is the smallest number you must add to 164 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
  13. 13.What is the remainder when 259 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
  14. 14.Which of these numbers is divisible by 6? (a) 1184 (b) 1176 (c) 1181 (d) 1173
  15. 15.Which of these numbers is divisible by 8? (a) 4630 (b) 4639 (c) 4618 (d) 4624
  16. 16.What is the smallest number you must add to 58 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  17. 17.Which of these numbers is divisible by 3? (a) 1613 (b) 1608 (c) 1606 (d) 1609
  18. 18.What is the remainder when 293 is divided by 6? (a) 4 (b) 1 (c) 5 (d) 3
  19. 19.What is the smallest number you must add to 292 to make it divisible by 6? (a) 1 (b) 2 (c) 4 (d) 3
  20. 20.Which of these numbers is divisible by 4? (a) 957 (b) 950 (c) 952 (d) 955
  21. 21.What is the remainder when 219 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 2
  22. 22.Which of these numbers is divisible by 9? (a) 963 (b) 962 (c) 966 (d) 974
  23. 23.Which of these numbers is divisible by 6? (a) 374 (b) 383 (c) 372 (d) 367
  24. 24.What is the units digit of 3^33? (a) 1 (b) 7 (c) 3 (d) 9
  25. 25.What is the smallest number you must add to 372 to make it divisible by 11? (a) 9 (b) 1 (c) 3 (d) 2
  26. 26.Which of these numbers is divisible by 4? (a) 2357 (b) 2360 (c) 2365 (d) 2363
  27. 27.Which of these numbers is divisible by 8? (a) 3396 (b) 3405 (c) 3415 (d) 3400
  28. 28.Which of these numbers is divisible by 6? (a) 909 (b) 897 (c) 904 (d) 900
  29. 29.Which of these numbers is divisible by 8? (a) 1790 (b) 1778 (c) 1776 (d) 1773
  30. 30.Which of these numbers is divisible by 9? (a) 7977 (b) 7968 (c) 7974 (d) 7989
  31. 31.What is the units digit of 2^30? (a) 6 (b) 4 (c) 2 (d) 8
  32. 32.What is the remainder when 718 is divided by 9? (a) 2 (b) 6 (c) 7 (d) 8
  33. 33.What is the units digit of 9^29? (a) 7 (b) 8 (c) 9 (d) 1
  34. 34.What is the smallest number you must add to 118 to make it divisible by 11? (a) 4 (b) 8 (c) 3 (d) 2
  35. 35.What is the units digit of 2^53? (a) 2 (b) 4 (c) 6 (d) 8
  36. 36.What is the smallest number you must add to 247 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
  37. 37.What is the units digit of 2^56? (a) 6 (b) 2 (c) 4 (d) 8
  38. 38.What is the smallest number you must add to 562 to make it divisible by 11? (a) 10 (b) 1 (c) 9 (d) 8
  39. 39.What is the units digit of 9^33? (a) 8 (b) 1 (c) 9 (d) 7
  40. 40.What is the smallest number you must add to 532 to make it divisible by 9? (a) 7 (b) 6 (c) 1 (d) 8

Answer key

  1. 3
  2. 4
  3. 7
  4. 2
  5. 5
  6. 3
  7. 2385
  8. 7
  9. 5
  10. 2
  11. 4
  12. 1
  13. 3
  14. 1176
  15. 4624
  16. 2
  17. 1608
  18. 5
  19. 2
  20. 952
  21. 3
  22. 963
  23. 372
  24. 3
  25. 2
  26. 2360
  27. 3400
  28. 900
  29. 1776
  30. 7974
  31. 4
  32. 7
  33. 9
  34. 3
  35. 2
  36. 3
  37. 6
  38. 10
  39. 9
  40. 8

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