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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 900 · 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 474 to make it divisible by 12? (a) 5 (b) 8 (c) 7 (d) 6
  2. 2.What is the units digit of 9^26? (a) 1 (b) 2 (c) 0 (d) 9
  3. 3.Which of these numbers is divisible by 11? (a) 9721 (b) 9696 (c) 9711 (d) 9702
  4. 4.Which of these numbers is divisible by 11? (a) 1398 (b) 1417 (c) 1426 (d) 1408
  5. 5.What is the smallest number you must add to 443 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 2
  6. 6.Which of these numbers is divisible by 11? (a) 4107 (b) 4092 (c) 4090 (d) 4096
  7. 7.What is the remainder when 398 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
  8. 8.What is the smallest number you must add to 724 to make it divisible by 11? (a) 3 (b) 2 (c) 9 (d) 1
  9. 9.What is the smallest number you must add to 467 to make it divisible by 9? (a) 8 (b) 3 (c) 2 (d) 1
  10. 10.Which of these numbers is divisible by 11? (a) 2318 (b) 2308 (c) 2329 (d) 2310
  11. 11.What is the remainder when 227 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
  12. 12.Which of these numbers is divisible by 9? (a) 4776 (b) 4769 (c) 4756 (d) 4761
  13. 13.What is the units digit of 3^11? (a) 7 (b) 9 (c) 1 (d) 3
  14. 14.What is the smallest number you must add to 389 to make it divisible by 9? (a) 2 (b) 7 (c) 6 (d) 8
  15. 15.What is the smallest number you must add to 644 to make it divisible by 9? (a) 5 (b) 6 (c) 3 (d) 4
  16. 16.Which of these numbers is divisible by 6? (a) 1551 (b) 1548 (c) 1556 (d) 1544
  17. 17.What is the smallest number you must add to 337 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 5
  18. 18.Which of these numbers is divisible by 3? (a) 2457 (b) 2461 (c) 2456 (d) 2458
  19. 19.What is the remainder when 877 is divided by 12? (a) 11 (b) 2 (c) 0 (d) 1
  20. 20.What is the remainder when 556 is divided by 7? (a) 2 (b) 4 (c) 5 (d) 3
  21. 21.What is the smallest number you must add to 136 to make it divisible by 9? (a) 8 (b) 1 (c) 7 (d) 6
  22. 22.What is the units digit of 4^14? (a) 4 (b) 5 (c) 7 (d) 6
  23. 23.What is the remainder when 235 is divided by 10? (a) 7 (b) 4 (c) 5 (d) 6
  24. 24.What is the smallest number you must add to 126 to make it divisible by 8? (a) 1 (b) 2 (c) 6 (d) 3
  25. 25.Which of these numbers is divisible by 9? (a) 6940 (b) 6932 (c) 6922 (d) 6930
  26. 26.Which of these numbers is divisible by 6? (a) 5256 (b) 5259 (c) 5254 (d) 5264
  27. 27.What is the smallest number you must add to 140 to make it divisible by 3? (a) 2 (b) 1
  28. 28.Which of these numbers is divisible by 3? (a) 2069 (b) 2074 (c) 2070 (d) 2072
  29. 29.What is the remainder when 474 is divided by 10? (a) 6 (b) 3 (c) 5 (d) 4
  30. 30.What is the remainder when 67 is divided by 3? (a) 0 (b) 1 (c) 2
  31. 31.What is the smallest number you must add to 542 to make it divisible by 11? (a) 3 (b) 7 (c) 8 (d) 9
  32. 32.Which of these numbers is divisible by 4? (a) 2667 (b) 2670 (c) 2673 (d) 2668
  33. 33.Which of these numbers is divisible by 6? (a) 440 (b) 431 (c) 433 (d) 432
  34. 34.Which of these numbers is divisible by 11? (a) 440 (b) 444 (c) 431 (d) 459
  35. 35.Which of these numbers is divisible by 3? (a) 439 (b) 438 (c) 442 (d) 436
  36. 36.What is the units digit of 2^22? (a) 4 (b) 6 (c) 8 (d) 2
  37. 37.What is the remainder when 247 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 4
  38. 38.What is the remainder when 277 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
  39. 39.Which of these numbers is divisible by 3? (a) 1004 (b) 1000 (c) 999 (d) 997
  40. 40.What is the remainder when 196 is divided by 3? (a) 1 (b) 2 (c) 0

Answer key

  1. 6
  2. 1
  3. 9702
  4. 1408
  5. 5
  6. 4092
  7. 3
  8. 2
  9. 1
  10. 2310
  11. 3
  12. 4761
  13. 7
  14. 7
  15. 4
  16. 1548
  17. 5
  18. 2457
  19. 1
  20. 3
  21. 8
  22. 6
  23. 5
  24. 2
  25. 6930
  26. 5256
  27. 1
  28. 2070
  29. 4
  30. 1
  31. 8
  32. 2668
  33. 432
  34. 440
  35. 438
  36. 4
  37. 4
  38. 2
  39. 999
  40. 1

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