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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 483 · 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 531 is divided by 12? (a) 9 (b) 3 (c) 4 (d) 2
  2. 2.What is the smallest number you must add to 81 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  3. 3.What is the remainder when 158 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
  4. 4.What is the smallest number you must add to 954 to make it divisible by 12? (a) 7 (b) 8 (c) 6 (d) 5
  5. 5.Which of these numbers is divisible by 9? (a) 7952 (b) 7960 (c) 7956 (d) 7970
  6. 6.What is the units digit of 3^38? (a) 1 (b) 3 (c) 9 (d) 7
  7. 7.What is the smallest number you must add to 263 to make it divisible by 10? (a) 7 (b) 6 (c) 3 (d) 8
  8. 8.What is the smallest number you must add to 179 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  9. 9.What is the smallest number you must add to 428 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
  10. 10.What is the smallest number you must add to 316 to make it divisible by 11? (a) 2 (b) 4 (c) 8 (d) 3
  11. 11.What is the units digit of 2^45? (a) 6 (b) 8 (c) 4 (d) 2
  12. 12.What is the remainder when 424 is divided by 11? (a) 5 (b) 6 (c) 7 (d) 8
  13. 13.What is the smallest number you must add to 191 to make it divisible by 3? (a) 1 (b) 2
  14. 14.What is the remainder when 901 is divided by 12? (a) 2 (b) 1 (c) 11 (d) 0
  15. 15.What is the remainder when 322 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
  16. 16.What is the units digit of 3^21? (a) 7 (b) 9 (c) 1 (d) 3
  17. 17.What is the remainder when 775 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 7
  18. 18.What is the remainder when 584 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 1
  19. 19.What is the remainder when 182 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
  20. 20.What is the units digit of 2^57? (a) 6 (b) 8 (c) 2 (d) 4
  21. 21.What is the smallest number you must add to 373 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 5
  22. 22.What is the smallest number you must add to 150 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 6
  23. 23.What is the units digit of 3^48? (a) 9 (b) 3 (c) 7 (d) 1
  24. 24.What is the smallest number you must add to 181 to make it divisible by 3? (a) 2 (b) 1
  25. 25.What is the remainder when 46 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
  26. 26.What is the units digit of 9^55? (a) 1 (b) 8 (c) 9 (d) 7
  27. 27.What is the units digit of 8^30? (a) 4 (b) 8 (c) 6 (d) 2
  28. 28.What is the remainder when 662 is divided by 9? (a) 6 (b) 4 (c) 5 (d) 7
  29. 29.What is the units digit of 2^28? (a) 8 (b) 4 (c) 2 (d) 6
  30. 30.What is the remainder when 419 is divided by 12? (a) 10 (b) 9 (c) 11 (d) 1
  31. 31.What is the units digit of 7^59? (a) 1 (b) 3 (c) 9 (d) 7
  32. 32.What is the units digit of 7^17? (a) 1 (b) 9 (c) 7 (d) 3
  33. 33.What is the units digit of 3^43? (a) 9 (b) 3 (c) 1 (d) 7
  34. 34.Which of these numbers is divisible by 11? (a) 7397 (b) 7408 (c) 7415 (d) 7403
  35. 35.Which of these numbers is divisible by 4? (a) 2116 (b) 2118 (c) 2115 (d) 2121
  36. 36.What is the units digit of 9^54? (a) 0 (b) 2 (c) 1 (d) 9
  37. 37.What is the remainder when 368 is divided by 9? (a) 6 (b) 7 (c) 8 (d) 1
  38. 38.What is the smallest number you must add to 882 to make it divisible by 12? (a) 5 (b) 7 (c) 6 (d) 8
  39. 39.What is the smallest number you must add to 424 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 4
  40. 40.What is the units digit of 9^18? (a) 9 (b) 2 (c) 0 (d) 1

Answer key

  1. 3
  2. 3
  3. 2
  4. 6
  5. 7956
  6. 9
  7. 7
  8. 1
  9. 4
  10. 3
  11. 2
  12. 6
  13. 1
  14. 1
  15. 2
  16. 3
  17. 5
  18. 8
  19. 2
  20. 2
  21. 5
  22. 2
  23. 1
  24. 2
  25. 2
  26. 9
  27. 4
  28. 5
  29. 6
  30. 11
  31. 3
  32. 7
  33. 7
  34. 7403
  35. 2116
  36. 1
  37. 8
  38. 6
  39. 2
  40. 1

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