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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 563 · 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 330 to make it divisible by 8? (a) 7 (b) 6 (c) 5 (d) 2
  2. 2.What is the smallest number you must add to 273 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  3. 3.Which of these numbers is divisible by 11? (a) 4653 (b) 4651 (c) 4659 (d) 4670
  4. 4.What is the remainder when 88 is divided by 3? (a) 0 (b) 1 (c) 2
  5. 5.What is the remainder when 674 is divided by 9? (a) 1 (b) 8 (c) 7 (d) 6
  6. 6.Which of these numbers is divisible by 4? (a) 2170 (b) 2173 (c) 2167 (d) 2168
  7. 7.Which of these numbers is divisible by 3? (a) 2027 (b) 2028 (c) 2032 (d) 2029
  8. 8.What is the smallest number you must add to 274 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  9. 9.Which of these numbers is divisible by 4? (a) 209 (b) 202 (c) 204 (d) 206
  10. 10.What is the smallest number you must add to 418 to make it divisible by 9? (a) 7 (b) 6 (c) 5 (d) 4
  11. 11.What is the remainder when 245 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
  12. 12.What is the smallest number you must add to 176 to make it divisible by 3? (a) 2 (b) 1
  13. 13.What is the smallest number you must add to 455 to make it divisible by 11? (a) 4 (b) 8 (c) 7 (d) 6
  14. 14.What is the units digit of 4^36? (a) 7 (b) 6 (c) 4 (d) 5
  15. 15.What is the remainder when 228 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
  16. 16.What is the remainder when 183 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
  17. 17.Which of these numbers is divisible by 9? (a) 1391 (b) 1395 (c) 1406 (d) 1396
  18. 18.What is the remainder when 203 is divided by 10? (a) 7 (b) 3 (c) 4 (d) 2
  19. 19.Which of these numbers is divisible by 3? (a) 2603 (b) 2605 (c) 2608 (d) 2604
  20. 20.What is the units digit of 3^11? (a) 7 (b) 1 (c) 9 (d) 3
  21. 21.Which of these numbers is divisible by 3? (a) 1348 (b) 1346 (c) 1347 (d) 1351
  22. 22.What is the units digit of 3^41? (a) 9 (b) 7 (c) 1 (d) 3
  23. 23.What is the units digit of 2^22? (a) 8 (b) 6 (c) 2 (d) 4
  24. 24.Which of these numbers is divisible by 11? (a) 8537 (b) 8521 (c) 8525 (d) 8526
  25. 25.What is the smallest number you must add to 187 to make it divisible by 10? (a) 7 (b) 2 (c) 4 (d) 3
  26. 26.What is the remainder when 117 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
  27. 27.What is the smallest number you must add to 253 to make it divisible by 7? (a) 1 (b) 6 (c) 5 (d) 4
  28. 28.Which of these numbers is divisible by 3? (a) 1199 (b) 1196 (c) 1197 (d) 1202
  29. 29.What is the smallest number you must add to 596 to make it divisible by 12? (a) 3 (b) 4 (c) 8 (d) 5
  30. 30.Which of these numbers is divisible by 6? (a) 4216 (b) 4206 (c) 4202 (d) 4207
  31. 31.What is the units digit of 2^52? (a) 6 (b) 2 (c) 8 (d) 4
  32. 32.What is the remainder when 174 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
  33. 33.What is the remainder when 286 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
  34. 34.What is the smallest number you must add to 636 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 2
  35. 35.Which of these numbers is divisible by 3? (a) 652 (b) 656 (c) 654 (d) 659
  36. 36.What is the smallest number you must add to 199 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  37. 37.What is the smallest number you must add to 444 to make it divisible by 9? (a) 7 (b) 6 (c) 5 (d) 3
  38. 38.Which of these numbers is divisible by 9? (a) 5841 (b) 5851 (c) 5842 (d) 5835
  39. 39.What is the remainder when 219 is divided by 10? (a) 1 (b) 8 (c) 9 (d) 7
  40. 40.What is the remainder when 145 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0

Answer key

  1. 6
  2. 3
  3. 4653
  4. 1
  5. 8
  6. 2168
  7. 2028
  8. 2
  9. 204
  10. 5
  11. 1
  12. 1
  13. 7
  14. 6
  15. 3
  16. 3
  17. 1395
  18. 3
  19. 2604
  20. 7
  21. 1347
  22. 3
  23. 4
  24. 8525
  25. 3
  26. 2
  27. 6
  28. 1197
  29. 4
  30. 4206
  31. 6
  32. 4
  33. 2
  34. 3
  35. 654
  36. 1
  37. 6
  38. 5841
  39. 9
  40. 1

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