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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 44 · 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.Which of these numbers is divisible by 6? (a) 2043 (b) 2047 (c) 2057 (d) 2046
  2. 2.What is the smallest number you must add to 208 to make it divisible by 3? (a) 2 (b) 1
  3. 3.Which of these numbers is divisible by 8? (a) 2039 (b) 2051 (c) 2040 (d) 2047
  4. 4.Which of these numbers is divisible by 11? (a) 9222 (b) 9248 (c) 9236 (d) 9229
  5. 5.What is the units digit of 2^48? (a) 4 (b) 6 (c) 2 (d) 8
  6. 6.What is the remainder when 233 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
  7. 7.What is the units digit of 7^60? (a) 7 (b) 9 (c) 3 (d) 1
  8. 8.What is the remainder when 637 is divided by 10? (a) 8 (b) 3 (c) 6 (d) 7
  9. 9.Which of these numbers is divisible by 4? (a) 1725 (b) 1730 (c) 1724 (d) 1723
  10. 10.Which of these numbers is divisible by 9? (a) 558 (b) 559 (c) 553 (d) 570
  11. 11.What is the units digit of 4^12? (a) 6 (b) 5 (c) 7 (d) 4
  12. 12.What is the units digit of 4^32? (a) 7 (b) 4 (c) 5 (d) 6
  13. 13.What is the remainder when 299 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
  14. 14.What is the smallest number you must add to 445 to make it divisible by 10? (a) 6 (b) 7 (c) 4 (d) 5
  15. 15.Which of these numbers is divisible by 6? (a) 857 (b) 865 (c) 860 (d) 858
  16. 16.What is the smallest number you must add to 460 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 1
  17. 17.Which of these numbers is divisible by 8? (a) 4943 (b) 4924 (c) 4931 (d) 4928
  18. 18.What is the remainder when 179 is divided by 3? (a) 0 (b) 2 (c) 1
  19. 19.What is the smallest number you must add to 101 to make it divisible by 3? (a) 2 (b) 1
  20. 20.What is the smallest number you must add to 404 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
  21. 21.What is the units digit of 3^59? (a) 9 (b) 3 (c) 1 (d) 7
  22. 22.Which of these numbers is divisible by 8? (a) 6078 (b) 6064 (c) 6062 (d) 6069
  23. 23.What is the units digit of 7^29? (a) 9 (b) 1 (c) 3 (d) 7
  24. 24.Which of these numbers is divisible by 3? (a) 1051 (b) 1054 (c) 1058 (d) 1053
  25. 25.What is the smallest number you must add to 250 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  26. 26.What is the smallest number you must add to 205 to make it divisible by 11? (a) 4 (b) 5 (c) 3 (d) 7
  27. 27.What is the smallest number you must add to 219 to make it divisible by 9? (a) 7 (b) 5 (c) 3 (d) 6
  28. 28.What is the smallest number you must add to 54 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  29. 29.Which of these numbers is divisible by 8? (a) 5418 (b) 5428 (c) 5435 (d) 5424
  30. 30.What is the remainder when 123 is divided by 12? (a) 9 (b) 2 (c) 3 (d) 4
  31. 31.What is the smallest number you must add to 181 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 4
  32. 32.What is the smallest number you must add to 565 to make it divisible by 7? (a) 1 (b) 2 (c) 3 (d) 5
  33. 33.Which of these numbers is divisible by 4? (a) 2980 (b) 2979 (c) 2983 (d) 2986
  34. 34.What is the remainder when 166 is divided by 11? (a) 0 (b) 2 (c) 10 (d) 1
  35. 35.Which of these numbers is divisible by 11? (a) 7818 (b) 7810 (c) 7805 (d) 7828
  36. 36.Which of these numbers is divisible by 4? (a) 1498 (b) 1500 (c) 1503 (d) 1507
  37. 37.Which of these numbers is divisible by 9? (a) 6118 (b) 6120 (c) 6134 (d) 6128
  38. 38.What is the smallest number you must add to 115 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 3
  39. 39.What is the smallest number you must add to 379 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 1
  40. 40.Which of these numbers is divisible by 4? (a) 1847 (b) 1853 (c) 1850 (d) 1848

Answer key

  1. 2046
  2. 2
  3. 2040
  4. 9229
  5. 6
  6. 1
  7. 1
  8. 7
  9. 1724
  10. 558
  11. 6
  12. 6
  13. 3
  14. 5
  15. 858
  16. 8
  17. 4928
  18. 2
  19. 1
  20. 4
  21. 7
  22. 6064
  23. 7
  24. 1053
  25. 2
  26. 4
  27. 6
  28. 2
  29. 5424
  30. 3
  31. 5
  32. 2
  33. 2980
  34. 1
  35. 7810
  36. 1500
  37. 6120
  38. 4
  39. 6
  40. 1848

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