← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 235 · 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) 2498 (b) 2492 (c) 2489 (d) 2493
  2. 2.What is the smallest number you must add to 219 to make it divisible by 8? (a) 4 (b) 6 (c) 3 (d) 5
  3. 3.What is the remainder when 302 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
  4. 4.What is the remainder when 315 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 2
  5. 5.Which of these numbers is divisible by 4? (a) 1158 (b) 1150 (c) 1152 (d) 1155
  6. 6.What is the units digit of 3^40? (a) 1 (b) 3 (c) 9 (d) 7
  7. 7.What is the remainder when 476 is divided by 8? (a) 5 (b) 6 (c) 3 (d) 4
  8. 8.What is the smallest number you must add to 313 to make it divisible by 7? (a) 3 (b) 5 (c) 2 (d) 1
  9. 9.What is the smallest number you must add to 639 to make it divisible by 12? (a) 10 (b) 3 (c) 9 (d) 8
  10. 10.What is the remainder when 145 is divided by 6? (a) 1 (b) 0 (c) 2 (d) 5
  11. 11.What is the smallest number you must add to 390 to make it divisible by 12? (a) 6 (b) 8 (c) 7 (d) 5
  12. 12.What is the remainder when 737 is divided by 10? (a) 3 (b) 8 (c) 6 (d) 7
  13. 13.What is the units digit of 3^46? (a) 3 (b) 7 (c) 1 (d) 9
  14. 14.Which of these numbers is divisible by 3? (a) 2209 (b) 2204 (c) 2206 (d) 2205
  15. 15.What is the units digit of 8^50? (a) 2 (b) 8 (c) 6 (d) 4
  16. 16.What is the smallest number you must add to 355 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
  17. 17.What is the smallest number you must add to 451 to make it divisible by 9? (a) 1 (b) 8 (c) 7 (d) 6
  18. 18.What is the remainder when 341 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
  19. 19.What is the smallest number you must add to 268 to make it divisible by 12? (a) 7 (b) 8 (c) 9 (d) 4
  20. 20.What is the units digit of 2^16? (a) 4 (b) 6 (c) 2 (d) 8
  21. 21.What is the smallest number you must add to 293 to make it divisible by 4? (a) 3 (b) 1 (c) 2
  22. 22.Which of these numbers is divisible by 4? (a) 3206 (b) 3199 (c) 3202 (d) 3200
  23. 23.Which of these numbers is divisible by 3? (a) 1955 (b) 1948 (c) 1950 (d) 1951
  24. 24.What is the units digit of 7^18? (a) 9 (b) 7 (c) 1 (d) 3
  25. 25.What is the remainder when 244 is divided by 10? (a) 6 (b) 5 (c) 4 (d) 3
  26. 26.What is the smallest number you must add to 440 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
  27. 27.Which of these numbers is divisible by 3? (a) 773 (b) 775 (c) 769 (d) 771
  28. 28.Which of these numbers is divisible by 6? (a) 991 (b) 985 (c) 997 (d) 990
  29. 29.What is the units digit of 9^13? (a) 9 (b) 8 (c) 7 (d) 1
  30. 30.What is the remainder when 34 is divided by 3? (a) 0 (b) 1 (c) 2
  31. 31.What is the smallest number you must add to 106 to make it divisible by 4? (a) 3 (b) 2 (c) 1
  32. 32.Which of these numbers is divisible by 3? (a) 1502 (b) 1500 (c) 1498 (d) 1504
  33. 33.What is the smallest number you must add to 803 to make it divisible by 12? (a) 2 (b) 1 (c) 11 (d) 3
  34. 34.What is the units digit of 7^53? (a) 3 (b) 9 (c) 7 (d) 1
  35. 35.Which of these numbers is divisible by 3? (a) 2620 (b) 2616 (c) 2618 (d) 2615
  36. 36.What is the units digit of 4^34? (a) 4 (b) 5 (c) 7 (d) 6
  37. 37.What is the units digit of 2^17? (a) 2 (b) 8 (c) 6 (d) 4
  38. 38.Which of these numbers is divisible by 11? (a) 5048 (b) 5049 (c) 5050 (d) 5067
  39. 39.What is the smallest number you must add to 604 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
  40. 40.What is the smallest number you must add to 314 to make it divisible by 7? (a) 1 (b) 6 (c) 2 (d) 3

Answer key

  1. 2492
  2. 5
  3. 2
  4. 3
  5. 1152
  6. 1
  7. 4
  8. 2
  9. 9
  10. 1
  11. 6
  12. 7
  13. 9
  14. 2205
  15. 4
  16. 5
  17. 8
  18. 1
  19. 8
  20. 6
  21. 3
  22. 3200
  23. 1950
  24. 9
  25. 4
  26. 4
  27. 771
  28. 990
  29. 9
  30. 1
  31. 2
  32. 1500
  33. 1
  34. 7
  35. 2616
  36. 6
  37. 2
  38. 5049
  39. 4
  40. 1

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad