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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 670 · 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 819 is divided by 11? (a) 6 (b) 5 (c) 4 (d) 7
  2. 2.What is the remainder when 103 is divided by 3? (a) 2 (b) 1 (c) 0
  3. 3.Which of these numbers is divisible by 9? (a) 2950 (b) 2941 (c) 2956 (d) 2943
  4. 4.What is the smallest number you must add to 236 to make it divisible by 11? (a) 8 (b) 6 (c) 5 (d) 7
  5. 5.What is the remainder when 193 is divided by 8? (a) 1 (b) 7 (c) 0 (d) 2
  6. 6.Which of these numbers is divisible by 9? (a) 7017 (b) 7011 (c) 7010 (d) 7026
  7. 7.What is the smallest number you must add to 439 to make it divisible by 10? (a) 1 (b) 9 (c) 3 (d) 2
  8. 8.Which of these numbers is divisible by 3? (a) 2344 (b) 2343 (c) 2341 (d) 2347
  9. 9.What is the smallest number you must add to 366 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
  10. 10.What is the units digit of 4^33? (a) 5 (b) 4 (c) 3 (d) 6
  11. 11.What is the units digit of 3^53? (a) 9 (b) 1 (c) 3 (d) 7
  12. 12.Which of these numbers is divisible by 9? (a) 1989 (b) 2002 (c) 1987 (d) 1991
  13. 13.Which of these numbers is divisible by 3? (a) 257 (b) 254 (c) 255 (d) 259
  14. 14.What is the remainder when 82 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
  15. 15.Which of these numbers is divisible by 9? (a) 6423 (b) 6413 (c) 6432 (d) 6417
  16. 16.What is the remainder when 416 is divided by 9? (a) 7 (b) 1 (c) 2 (d) 3
  17. 17.What is the remainder when 143 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
  18. 18.Which of these numbers is divisible by 6? (a) 3336 (b) 3337 (c) 3343 (d) 3331
  19. 19.What is the units digit of 8^42? (a) 6 (b) 4 (c) 8 (d) 2
  20. 20.What is the smallest number you must add to 501 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 3
  21. 21.What is the smallest number you must add to 371 to make it divisible by 8? (a) 4 (b) 5 (c) 6 (d) 3
  22. 22.Which of these numbers is divisible by 3? (a) 1523 (b) 1516 (c) 1518 (d) 1520
  23. 23.Which of these numbers is divisible by 3? (a) 615 (b) 617 (c) 620 (d) 614
  24. 24.What is the units digit of 2^28? (a) 2 (b) 4 (c) 8 (d) 6
  25. 25.What is the smallest number you must add to 147 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  26. 26.What is the smallest number you must add to 139 to make it divisible by 8? (a) 5 (b) 6 (c) 4 (d) 3
  27. 27.What is the smallest number you must add to 533 to make it divisible by 7? (a) 4 (b) 5 (c) 1 (d) 6
  28. 28.What is the units digit of 2^26? (a) 2 (b) 8 (c) 4 (d) 6
  29. 29.Which of these numbers is divisible by 4? (a) 3067 (b) 3064 (c) 3071 (d) 3062
  30. 30.What is the smallest number you must add to 623 to make it divisible by 10? (a) 3 (b) 8 (c) 6 (d) 7
  31. 31.What is the smallest number you must add to 233 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  32. 32.What is the smallest number you must add to 481 to make it divisible by 8? (a) 7 (b) 1 (c) 5 (d) 6
  33. 33.What is the units digit of 7^41? (a) 3 (b) 9 (c) 1 (d) 7
  34. 34.What is the smallest number you must add to 284 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
  35. 35.What is the units digit of 8^41? (a) 8 (b) 4 (c) 2 (d) 6
  36. 36.Which of these numbers is divisible by 9? (a) 1387 (b) 1383 (c) 1386 (d) 1403
  37. 37.What is the smallest number you must add to 813 to make it divisible by 11? (a) 2 (b) 10 (c) 1 (d) 3
  38. 38.What is the smallest number you must add to 120 to make it divisible by 9? (a) 7 (b) 5 (c) 3 (d) 6
  39. 39.Which of these numbers is divisible by 4? (a) 3351 (b) 3348 (c) 3354 (d) 3345
  40. 40.What is the units digit of 7^24? (a) 3 (b) 7 (c) 9 (d) 1

Answer key

  1. 5
  2. 1
  3. 2943
  4. 6
  5. 1
  6. 7011
  7. 1
  8. 2343
  9. 4
  10. 4
  11. 3
  12. 1989
  13. 255
  14. 2
  15. 6417
  16. 2
  17. 3
  18. 3336
  19. 4
  20. 3
  21. 5
  22. 1518
  23. 615
  24. 6
  25. 1
  26. 5
  27. 6
  28. 4
  29. 3064
  30. 7
  31. 3
  32. 7
  33. 7
  34. 1
  35. 8
  36. 1386
  37. 1
  38. 6
  39. 3348
  40. 1

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