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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 810 · 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 4? (a) 685 (b) 681 (c) 684 (d) 690
  2. 2.What is the smallest number you must add to 304 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  3. 3.What is the smallest number you must add to 431 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 2
  4. 4.What is the units digit of 7^12? (a) 3 (b) 7 (c) 9 (d) 1
  5. 5.Which of these numbers is divisible by 9? (a) 5767 (b) 5760 (c) 5770 (d) 5759
  6. 6.Which of these numbers is divisible by 6? (a) 4304 (b) 4295 (c) 4301 (d) 4296
  7. 7.Which of these numbers is divisible by 4? (a) 2645 (b) 2640 (c) 2639 (d) 2643
  8. 8.What is the smallest number you must add to 264 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
  9. 9.What is the smallest number you must add to 147 to make it divisible by 12? (a) 8 (b) 3 (c) 10 (d) 9
  10. 10.What is the units digit of 7^27? (a) 9 (b) 1 (c) 7 (d) 3
  11. 11.Which of these numbers is divisible by 9? (a) 8044 (b) 8060 (c) 8047 (d) 8046
  12. 12.What is the remainder when 122 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
  13. 13.What is the remainder when 126 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
  14. 14.What is the remainder when 275 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 5
  15. 15.What is the remainder when 430 is divided by 12? (a) 11 (b) 9 (c) 2 (d) 10
  16. 16.What is the smallest number you must add to 367 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 5
  17. 17.What is the smallest number you must add to 454 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 4
  18. 18.Which of these numbers is divisible by 9? (a) 5146 (b) 5148 (c) 5163 (d) 5151
  19. 19.Which of these numbers is divisible by 4? (a) 1403 (b) 1395 (c) 1398 (d) 1396
  20. 20.Which of these numbers is divisible by 6? (a) 342 (b) 353 (c) 345 (d) 341
  21. 21.What is the remainder when 322 is divided by 11? (a) 3 (b) 8 (c) 4 (d) 2
  22. 22.Which of these numbers is divisible by 3? (a) 538 (b) 533 (c) 534 (d) 535
  23. 23.What is the units digit of 7^11? (a) 9 (b) 3 (c) 1 (d) 7
  24. 24.What is the remainder when 676 is divided by 11? (a) 7 (b) 5 (c) 6 (d) 4
  25. 25.Which of these numbers is divisible by 9? (a) 4782 (b) 4770 (c) 4762 (d) 4774
  26. 26.What is the units digit of 7^36? (a) 7 (b) 1 (c) 9 (d) 3
  27. 27.What is the smallest number you must add to 634 to make it divisible by 10? (a) 6 (b) 5 (c) 4 (d) 7
  28. 28.What is the smallest number you must add to 190 to make it divisible by 3? (a) 1 (b) 2
  29. 29.Which of these numbers is divisible by 4? (a) 1994 (b) 2001 (c) 1998 (d) 1996
  30. 30.What is the units digit of 8^58? (a) 4 (b) 8 (c) 6 (d) 2
  31. 31.What is the smallest number you must add to 243 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
  32. 32.What is the remainder when 179 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
  33. 33.What is the remainder when 174 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
  34. 34.What is the units digit of 7^59? (a) 1 (b) 9 (c) 3 (d) 7
  35. 35.What is the smallest number you must add to 251 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  36. 36.Which of these numbers is divisible by 6? (a) 2782 (b) 2778 (c) 2786 (d) 2777
  37. 37.What is the smallest number you must add to 416 to make it divisible by 7? (a) 4 (b) 6 (c) 3 (d) 5
  38. 38.What is the smallest number you must add to 433 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 4
  39. 39.What is the units digit of 3^25? (a) 3 (b) 7 (c) 1 (d) 9
  40. 40.What is the smallest number you must add to 388 to make it divisible by 10? (a) 8 (b) 2 (c) 3 (d) 1

Answer key

  1. 684
  2. 1
  3. 9
  4. 1
  5. 5760
  6. 4296
  7. 2640
  8. 1
  9. 9
  10. 3
  11. 8046
  12. 2
  13. 2
  14. 5
  15. 10
  16. 5
  17. 2
  18. 5148
  19. 1396
  20. 342
  21. 3
  22. 534
  23. 3
  24. 5
  25. 4770
  26. 1
  27. 6
  28. 2
  29. 1996
  30. 4
  31. 2
  32. 4
  33. 6
  34. 3
  35. 1
  36. 2778
  37. 4
  38. 5
  39. 3
  40. 2

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