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Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 517 · 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 357 to make it divisible by 9? (a) 2 (b) 6 (c) 4 (d) 3
  2. 2.What is the units digit of 2^38? (a) 2 (b) 4 (c) 8 (d) 6
  3. 3.What is the smallest number you must add to 149 to make it divisible by 9? (a) 4 (b) 3 (c) 5 (d) 6
  4. 4.What is the smallest number you must add to 182 to make it divisible by 12? (a) 11 (b) 2 (c) 9 (d) 10
  5. 5.What is the remainder when 172 is divided by 9? (a) 8 (b) 2 (c) 0 (d) 1
  6. 6.Which of these numbers is divisible by 9? (a) 2425 (b) 2415 (c) 2421 (d) 2432
  7. 7.What is the smallest number you must add to 161 to make it divisible by 4? (a) 2 (b) 3 (c) 1
  8. 8.Which of these numbers is divisible by 8? (a) 6340 (b) 6333 (c) 6346 (d) 6336
  9. 9.What is the smallest number you must add to 78 to make it divisible by 4? (a) 1 (b) 3 (c) 2
  10. 10.What is the units digit of 7^21? (a) 9 (b) 7 (c) 3 (d) 1
  11. 11.Which of these numbers is divisible by 9? (a) 591 (b) 583 (c) 585 (d) 595
  12. 12.What is the units digit of 8^57? (a) 8 (b) 4 (c) 6 (d) 2
  13. 13.What is the smallest number you must add to 725 to make it divisible by 10? (a) 6 (b) 5 (c) 7 (d) 4
  14. 14.What is the smallest number you must add to 373 to make it divisible by 7? (a) 5 (b) 4 (c) 2 (d) 6
  15. 15.What is the units digit of 8^36? (a) 8 (b) 6 (c) 2 (d) 4
  16. 16.What is the remainder when 114 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
  17. 17.Which of these numbers is divisible by 4? (a) 2072 (b) 2069 (c) 2075 (d) 2078
  18. 18.What is the remainder when 676 is divided by 10? (a) 6 (b) 5 (c) 7 (d) 4
  19. 19.What is the units digit of 4^23? (a) 3 (b) 5 (c) 6 (d) 4
  20. 20.What is the units digit of 9^58? (a) 1 (b) 9 (c) 0 (d) 2
  21. 21.What is the smallest number you must add to 225 to make it divisible by 8? (a) 1 (b) 7 (c) 5 (d) 6
  22. 22.Which of these numbers is divisible by 3? (a) 700 (b) 697 (c) 703 (d) 699
  23. 23.What is the remainder when 155 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
  24. 24.Which of these numbers is divisible by 4? (a) 1270 (b) 1268 (c) 1266 (d) 1275
  25. 25.What is the remainder when 782 is divided by 10? (a) 3 (b) 1 (c) 8 (d) 2
  26. 26.What is the smallest number you must add to 465 to make it divisible by 11? (a) 9 (b) 8 (c) 3 (d) 7
  27. 27.What is the units digit of 8^16? (a) 2 (b) 4 (c) 6 (d) 8
  28. 28.What is the units digit of 3^40? (a) 9 (b) 3 (c) 7 (d) 1
  29. 29.Which of these numbers is divisible by 3? (a) 799 (b) 797 (c) 798 (d) 802
  30. 30.What is the units digit of 8^23? (a) 6 (b) 4 (c) 2 (d) 8
  31. 31.Which of these numbers is divisible by 3? (a) 1951 (b) 1955 (c) 1948 (d) 1950
  32. 32.What is the remainder when 44 is divided by 3? (a) 1 (b) 0 (c) 2
  33. 33.What is the smallest number you must add to 143 to make it divisible by 6? (a) 1 (b) 2 (c) 5 (d) 3
  34. 34.What is the units digit of 8^38? (a) 8 (b) 6 (c) 2 (d) 4
  35. 35.What is the remainder when 79 is divided by 3? (a) 2 (b) 0 (c) 1
  36. 36.Which of these numbers is divisible by 6? (a) 268 (b) 270 (c) 271 (d) 281
  37. 37.What is the smallest number you must add to 276 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
  38. 38.What is the remainder when 417 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 3
  39. 39.What is the remainder when 188 is divided by 10? (a) 7 (b) 2 (c) 9 (d) 8
  40. 40.Which of these numbers is divisible by 6? (a) 551 (b) 554 (c) 560 (d) 552

Answer key

  1. 3
  2. 4
  3. 4
  4. 10
  5. 1
  6. 2421
  7. 3
  8. 6336
  9. 2
  10. 7
  11. 585
  12. 8
  13. 5
  14. 5
  15. 6
  16. 4
  17. 2072
  18. 6
  19. 4
  20. 1
  21. 7
  22. 699
  23. 1
  24. 1268
  25. 2
  26. 8
  27. 6
  28. 1
  29. 798
  30. 2
  31. 1950
  32. 2
  33. 1
  34. 4
  35. 1
  36. 270
  37. 4
  38. 4
  39. 8
  40. 552

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