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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 974 · 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 units digit of 4^29? (a) 5 (b) 4 (c) 6 (d) 3
  2. 2.What is the smallest number you must add to 254 to make it divisible by 8? (a) 3 (b) 1 (c) 2 (d) 6
  3. 3.What is the smallest number you must add to 484 to make it divisible by 6? (a) 2 (b) 1 (c) 4 (d) 3
  4. 4.What is the remainder when 298 is divided by 7? (a) 5 (b) 6 (c) 3 (d) 4
  5. 5.Which of these numbers is divisible by 6? (a) 5105 (b) 5096 (c) 5092 (d) 5094
  6. 6.What is the remainder when 69 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
  7. 7.What is the smallest number you must add to 149 to make it divisible by 10? (a) 2 (b) 3 (c) 9 (d) 1
  8. 8.Which of these numbers is divisible by 4? (a) 1479 (b) 1483 (c) 1480 (d) 1486
  9. 9.What is the remainder when 172 is divided by 3? (a) 0 (b) 1 (c) 2
  10. 10.What is the remainder when 120 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
  11. 11.Which of these numbers is divisible by 11? (a) 8457 (b) 8429 (c) 8444 (d) 8437
  12. 12.What is the units digit of 7^34? (a) 7 (b) 3 (c) 9 (d) 1
  13. 13.What is the smallest number you must add to 223 to make it divisible by 7? (a) 1 (b) 2 (c) 6 (d) 3
  14. 14.What is the units digit of 3^52? (a) 9 (b) 7 (c) 1 (d) 3
  15. 15.Which of these numbers is divisible by 3? (a) 2635 (b) 2632 (c) 2634 (d) 2638
  16. 16.What is the units digit of 3^59? (a) 9 (b) 1 (c) 3 (d) 7
  17. 17.What is the units digit of 8^14? (a) 6 (b) 4 (c) 8 (d) 2
  18. 18.What is the units digit of 9^35? (a) 9 (b) 1 (c) 7 (d) 8
  19. 19.What is the units digit of 3^23? (a) 7 (b) 1 (c) 3 (d) 9
  20. 20.Which of these numbers is divisible by 3? (a) 2573 (b) 2569 (c) 2575 (d) 2571
  21. 21.What is the smallest number you must add to 265 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  22. 22.What is the remainder when 52 is divided by 3? (a) 1 (b) 2 (c) 0
  23. 23.What is the smallest number you must add to 665 to make it divisible by 10? (a) 5 (b) 7 (c) 6 (d) 4
  24. 24.What is the smallest number you must add to 356 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  25. 25.Which of these numbers is divisible by 11? (a) 4470 (b) 4484 (c) 4458 (d) 4466
  26. 26.What is the remainder when 354 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
  27. 27.What is the smallest number you must add to 41 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  28. 28.Which of these numbers is divisible by 4? (a) 587 (b) 593 (c) 589 (d) 588
  29. 29.What is the remainder when 263 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 1
  30. 30.What is the smallest number you must add to 131 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
  31. 31.What is the smallest number you must add to 233 to make it divisible by 6? (a) 1 (b) 5 (c) 2 (d) 3
  32. 32.Which of these numbers is divisible by 9? (a) 1346 (b) 1350 (c) 1360 (d) 1351
  33. 33.Which of these numbers is divisible by 11? (a) 1419 (b) 1431 (c) 1423 (d) 1413
  34. 34.Which of these numbers is divisible by 6? (a) 977 (b) 971 (c) 981 (d) 972
  35. 35.Which of these numbers is divisible by 9? (a) 4722 (b) 4726 (c) 4725 (d) 4742
  36. 36.Which of these numbers is divisible by 6? (a) 2114 (b) 2106 (c) 2107 (d) 2104
  37. 37.Which of these numbers is divisible by 11? (a) 7138 (b) 7155 (c) 7142 (d) 7139
  38. 38.What is the smallest number you must add to 142 to make it divisible by 3? (a) 1 (b) 2
  39. 39.What is the remainder when 794 is divided by 10? (a) 3 (b) 4 (c) 5 (d) 6
  40. 40.What is the smallest number you must add to 149 to make it divisible by 12? (a) 8 (b) 7 (c) 5 (d) 6

Answer key

  1. 4
  2. 2
  3. 2
  4. 4
  5. 5094
  6. 3
  7. 1
  8. 1480
  9. 1
  10. 1
  11. 8437
  12. 9
  13. 1
  14. 1
  15. 2634
  16. 7
  17. 4
  18. 9
  19. 7
  20. 2571
  21. 3
  22. 1
  23. 5
  24. 4
  25. 4466
  26. 4
  27. 3
  28. 588
  29. 5
  30. 4
  31. 1
  32. 1350
  33. 1419
  34. 972
  35. 4725
  36. 2106
  37. 7139
  38. 2
  39. 4
  40. 7

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