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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 781 · 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 remainder when 268 is divided by 8? (a) 3 (b) 5 (c) 6 (d) 4
  2. 2.What is the remainder when 627 is divided by 10? (a) 8 (b) 3 (c) 6 (d) 7
  3. 3.What is the remainder when 43 is divided by 3? (a) 1 (b) 0 (c) 2
  4. 4.What is the remainder when 173 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
  5. 5.What is the remainder when 163 is divided by 3? (a) 2 (b) 0 (c) 1
  6. 6.What is the units digit of 2^15? (a) 8 (b) 2 (c) 4 (d) 6
  7. 7.What is the smallest number you must add to 316 to make it divisible by 10? (a) 6 (b) 5 (c) 3 (d) 4
  8. 8.What is the smallest number you must add to 241 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  9. 9.Which of these numbers is divisible by 3? (a) 1367 (b) 1363 (c) 1360 (d) 1362
  10. 10.Which of these numbers is divisible by 4? (a) 2041 (b) 2037 (c) 2036 (d) 2034
  11. 11.Which of these numbers is divisible by 8? (a) 1057 (b) 1052 (c) 1065 (d) 1056
  12. 12.Which of these numbers is divisible by 3? (a) 1409 (b) 1410 (c) 1411 (d) 1414
  13. 13.What is the remainder when 262 is divided by 7? (a) 4 (b) 3 (c) 2 (d) 5
  14. 14.What is the smallest number you must add to 199 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
  15. 15.What is the remainder when 197 is divided by 8? (a) 5 (b) 3 (c) 6 (d) 4
  16. 16.Which of these numbers is divisible by 6? (a) 1311 (b) 1318 (c) 1321 (d) 1314
  17. 17.What is the units digit of 8^12? (a) 8 (b) 4 (c) 6 (d) 2
  18. 18.What is the remainder when 597 is divided by 11? (a) 4 (b) 2 (c) 8 (d) 3
  19. 19.What is the remainder when 178 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
  20. 20.Which of these numbers is divisible by 9? (a) 668 (b) 679 (c) 666 (d) 663
  21. 21.What is the units digit of 7^52? (a) 1 (b) 3 (c) 9 (d) 7
  22. 22.What is the smallest number you must add to 467 to make it divisible by 10? (a) 7 (b) 4 (c) 2 (d) 3
  23. 23.What is the smallest number you must add to 35 to make it divisible by 3? (a) 2 (b) 1
  24. 24.Which of these numbers is divisible by 3? (a) 1709 (b) 1702 (c) 1705 (d) 1704
  25. 25.What is the smallest number you must add to 191 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  26. 26.Which of these numbers is divisible by 4? (a) 910 (b) 914 (c) 912 (d) 917
  27. 27.What is the units digit of 9^54? (a) 9 (b) 1 (c) 2 (d) 0
  28. 28.What is the units digit of 8^19? (a) 4 (b) 6 (c) 2 (d) 8
  29. 29.What is the units digit of 8^42? (a) 2 (b) 6 (c) 8 (d) 4
  30. 30.What is the smallest number you must add to 605 to make it divisible by 9? (a) 7 (b) 2 (c) 6 (d) 8
  31. 31.What is the remainder when 478 is divided by 8? (a) 7 (b) 6 (c) 2 (d) 5
  32. 32.What is the remainder when 65 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
  33. 33.What is the units digit of 9^47? (a) 1 (b) 8 (c) 7 (d) 9
  34. 34.Which of these numbers is divisible by 11? (a) 6858 (b) 6871 (c) 6853 (d) 6846
  35. 35.What is the smallest number you must add to 383 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
  36. 36.What is the smallest number you must add to 451 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 1
  37. 37.Which of these numbers is divisible by 6? (a) 4380 (b) 4385 (c) 4377 (d) 4391
  38. 38.What is the remainder when 286 is divided by 7? (a) 1 (b) 5 (c) 6 (d) 4
  39. 39.What is the units digit of 7^40? (a) 9 (b) 3 (c) 7 (d) 1
  40. 40.What is the smallest number you must add to 196 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 6

Answer key

  1. 4
  2. 7
  3. 1
  4. 1
  5. 1
  6. 8
  7. 4
  8. 3
  9. 1362
  10. 2036
  11. 1056
  12. 1410
  13. 3
  14. 1
  15. 5
  16. 1314
  17. 6
  18. 3
  19. 3
  20. 666
  21. 1
  22. 3
  23. 1
  24. 1704
  25. 1
  26. 912
  27. 1
  28. 2
  29. 4
  30. 7
  31. 6
  32. 1
  33. 9
  34. 6853
  35. 2
  36. 8
  37. 4380
  38. 6
  39. 1
  40. 4

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