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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 972 · 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 193 to make it divisible by 7? (a) 5 (b) 4 (c) 3 (d) 2
  2. 2.What is the remainder when 293 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
  3. 3.What is the smallest number you must add to 109 to make it divisible by 3? (a) 1 (b) 2
  4. 4.Which of these numbers is divisible by 6? (a) 950 (b) 954 (c) 957 (d) 961
  5. 5.What is the smallest number you must add to 223 to make it divisible by 4? (a) 2 (b) 1 (c) 3
  6. 6.What is the remainder when 476 is divided by 9? (a) 7 (b) 6 (c) 8 (d) 1
  7. 7.What is the smallest number you must add to 321 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
  8. 8.What is the remainder when 130 is divided by 3? (a) 1 (b) 2 (c) 0
  9. 9.What is the remainder when 453 is divided by 11? (a) 9 (b) 3 (c) 2 (d) 1
  10. 10.What is the smallest number you must add to 547 to make it divisible by 10? (a) 4 (b) 2 (c) 3 (d) 7
  11. 11.What is the smallest number you must add to 275 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  12. 12.What is the units digit of 4^38? (a) 7 (b) 6 (c) 4 (d) 5
  13. 13.What is the units digit of 3^57? (a) 9 (b) 7 (c) 1 (d) 3
  14. 14.Which of these numbers is divisible by 8? (a) 1539 (b) 1536 (c) 1551 (d) 1534
  15. 15.What is the smallest number you must add to 94 to make it divisible by 7? (a) 6 (b) 3 (c) 4 (d) 5
  16. 16.What is the units digit of 8^45? (a) 6 (b) 8 (c) 2 (d) 4
  17. 17.What is the smallest number you must add to 428 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
  18. 18.Which of these numbers is divisible by 6? (a) 2477 (b) 2481 (c) 2470 (d) 2472
  19. 19.What is the smallest number you must add to 597 to make it divisible by 9? (a) 5 (b) 7 (c) 6 (d) 3
  20. 20.What is the smallest number you must add to 223 to make it divisible by 3? (a) 1 (b) 2
  21. 21.Which of these numbers is divisible by 6? (a) 4057 (b) 4052 (c) 4049 (d) 4050
  22. 22.What is the remainder when 524 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
  23. 23.Which of these numbers is divisible by 3? (a) 2435 (b) 2429 (c) 2432 (d) 2430
  24. 24.Which of these numbers is divisible by 3? (a) 1117 (b) 1124 (c) 1119 (d) 1121
  25. 25.Which of these numbers is divisible by 11? (a) 3889 (b) 3902 (c) 3912 (d) 3894
  26. 26.What is the remainder when 332 is divided by 10? (a) 2 (b) 3 (c) 1 (d) 8
  27. 27.What is the units digit of 3^18? (a) 1 (b) 7 (c) 9 (d) 3
  28. 28.What is the remainder when 93 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
  29. 29.What is the units digit of 2^29? (a) 6 (b) 2 (c) 4 (d) 8
  30. 30.What is the smallest number you must add to 142 to make it divisible by 3? (a) 1 (b) 2
  31. 31.Which of these numbers is divisible by 6? (a) 4389 (b) 4392 (c) 4397 (d) 4400
  32. 32.Which of these numbers is divisible by 9? (a) 4763 (b) 4761 (c) 4753 (d) 4771
  33. 33.Which of these numbers is divisible by 11? (a) 1643 (b) 1630 (c) 1652 (d) 1639
  34. 34.Which of these numbers is divisible by 6? (a) 3753 (b) 3756 (c) 3761 (d) 3763
  35. 35.What is the remainder when 113 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
  36. 36.Which of these numbers is divisible by 4? (a) 3149 (b) 3143 (c) 3147 (d) 3144
  37. 37.Which of these numbers is divisible by 6? (a) 4229 (b) 4238 (c) 4234 (d) 4230
  38. 38.What is the smallest number you must add to 444 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 7
  39. 39.Which of these numbers is divisible by 4? (a) 1912 (b) 1913 (c) 1910 (d) 1918
  40. 40.What is the remainder when 186 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2

Answer key

  1. 3
  2. 1
  3. 2
  4. 954
  5. 1
  6. 8
  7. 3
  8. 1
  9. 2
  10. 3
  11. 1
  12. 6
  13. 3
  14. 1536
  15. 4
  16. 8
  17. 4
  18. 2472
  19. 6
  20. 2
  21. 4050
  22. 4
  23. 2430
  24. 1119
  25. 3894
  26. 2
  27. 9
  28. 1
  29. 2
  30. 2
  31. 4392
  32. 4761
  33. 1639
  34. 3756
  35. 1
  36. 3144
  37. 4230
  38. 6
  39. 1912
  40. 2

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