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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 326 · 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 422 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
  2. 2.Which of these numbers is divisible by 4? (a) 1105 (b) 1099 (c) 1100 (d) 1102
  3. 3.What is the units digit of 9^52? (a) 9 (b) 0 (c) 1 (d) 2
  4. 4.What is the smallest number you must add to 645 to make it divisible by 8? (a) 4 (b) 5 (c) 2 (d) 3
  5. 5.What is the remainder when 358 is divided by 11? (a) 8 (b) 6 (c) 5 (d) 7
  6. 6.What is the units digit of 2^35? (a) 6 (b) 2 (c) 8 (d) 4
  7. 7.What is the smallest number you must add to 468 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 6
  8. 8.Which of these numbers is divisible by 6? (a) 1264 (b) 1259 (c) 1254 (d) 1250
  9. 9.What is the smallest number you must add to 163 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  10. 10.What is the units digit of 9^57? (a) 8 (b) 7 (c) 1 (d) 9
  11. 11.What is the remainder when 130 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 3
  12. 12.What is the remainder when 205 is divided by 11? (a) 6 (b) 4 (c) 7 (d) 8
  13. 13.Which of these numbers is divisible by 9? (a) 1376 (b) 1354 (c) 1365 (d) 1359
  14. 14.What is the smallest number you must add to 46 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  15. 15.What is the remainder when 95 is divided by 3? (a) 2 (b) 0 (c) 1
  16. 16.What is the remainder when 888 is divided by 11? (a) 7 (b) 8 (c) 9 (d) 3
  17. 17.Which of these numbers is divisible by 3? (a) 1355 (b) 1349 (c) 1351 (d) 1350
  18. 18.What is the remainder when 151 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
  19. 19.Which of these numbers is divisible by 8? (a) 2053 (b) 2042 (c) 2039 (d) 2040
  20. 20.Which of these numbers is divisible by 3? (a) 2525 (b) 2522 (c) 2527 (d) 2523
  21. 21.What is the smallest number you must add to 135 to make it divisible by 10? (a) 4 (b) 7 (c) 5 (d) 6
  22. 22.What is the smallest number you must add to 285 to make it divisible by 7? (a) 3 (b) 2 (c) 5 (d) 1
  23. 23.What is the remainder when 517 is divided by 12? (a) 11 (b) 1 (c) 2 (d) 0
  24. 24.What is the smallest number you must add to 67 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  25. 25.Which of these numbers is divisible by 8? (a) 5328 (b) 5323 (c) 5329 (d) 5343
  26. 26.What is the units digit of 3^26? (a) 1 (b) 7 (c) 3 (d) 9
  27. 27.Which of these numbers is divisible by 4? (a) 1798 (b) 1807 (c) 1801 (d) 1800
  28. 28.Which of these numbers is divisible by 6? (a) 1902 (b) 1906 (c) 1897 (d) 1911
  29. 29.Which of these numbers is divisible by 4? (a) 2316 (b) 2314 (c) 2319 (d) 2321
  30. 30.Which of these numbers is divisible by 11? (a) 3927 (b) 3919 (c) 3931 (d) 3948
  31. 31.What is the units digit of 3^43? (a) 7 (b) 3 (c) 1 (d) 9
  32. 32.What is the units digit of 9^56? (a) 1 (b) 9 (c) 2 (d) 0
  33. 33.Which of these numbers is divisible by 11? (a) 4561 (b) 4577 (c) 4565 (d) 4570
  34. 34.What is the smallest number you must add to 257 to make it divisible by 7? (a) 3 (b) 5 (c) 1 (d) 2
  35. 35.What is the smallest number you must add to 592 to make it divisible by 10? (a) 2 (b) 7 (c) 8 (d) 9
  36. 36.What is the remainder when 136 is divided by 9? (a) 1 (b) 0 (c) 8 (d) 2
  37. 37.Which of these numbers is divisible by 4? (a) 1620 (b) 1617 (c) 1623 (d) 1626
  38. 38.Which of these numbers is divisible by 4? (a) 1157 (b) 1153 (c) 1156 (d) 1163
  39. 39.What is the smallest number you must add to 77 to make it divisible by 3? (a) 1 (b) 2
  40. 40.What is the smallest number you must add to 345 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 1

Answer key

  1. 4
  2. 1100
  3. 1
  4. 3
  5. 6
  6. 8
  7. 1
  8. 1254
  9. 1
  10. 9
  11. 4
  12. 7
  13. 1359
  14. 2
  15. 2
  16. 8
  17. 1350
  18. 3
  19. 2040
  20. 2523
  21. 5
  22. 2
  23. 1
  24. 1
  25. 5328
  26. 9
  27. 1800
  28. 1902
  29. 2316
  30. 3927
  31. 7
  32. 1
  33. 4565
  34. 2
  35. 8
  36. 1
  37. 1620
  38. 1156
  39. 1
  40. 7

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