← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 969 · 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 208 is divided by 10? (a) 2 (b) 9 (c) 8 (d) 7
  2. 2.Which of these numbers is divisible by 11? (a) 5799 (b) 5797 (c) 5817 (d) 5796
  3. 3.Which of these numbers is divisible by 3? (a) 1119 (b) 1124 (c) 1120 (d) 1117
  4. 4.Which of these numbers is divisible by 4? (a) 1396 (b) 1394 (c) 1398 (d) 1403
  5. 5.What is the remainder when 38 is divided by 3? (a) 1 (b) 2 (c) 0
  6. 6.Which of these numbers is divisible by 6? (a) 1233 (b) 1227 (c) 1219 (d) 1224
  7. 7.What is the smallest number you must add to 330 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 2
  8. 8.What is the smallest number you must add to 519 to make it divisible by 8? (a) 1 (b) 7 (c) 2 (d) 3
  9. 9.What is the remainder when 248 is divided by 7? (a) 5 (b) 4 (c) 2 (d) 3
  10. 10.What is the smallest number you must add to 521 to make it divisible by 9? (a) 2 (b) 8 (c) 3 (d) 1
  11. 11.What is the smallest number you must add to 463 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 1
  12. 12.What is the units digit of 9^51? (a) 8 (b) 9 (c) 7 (d) 1
  13. 13.Which of these numbers is divisible by 9? (a) 6697 (b) 6705 (c) 6707 (d) 6721
  14. 14.Which of these numbers is divisible by 3? (a) 1526 (b) 1524 (c) 1523 (d) 1528
  15. 15.What is the smallest number you must add to 488 to make it divisible by 10? (a) 8 (b) 3 (c) 1 (d) 2
  16. 16.What is the units digit of 4^19? (a) 6 (b) 4 (c) 3 (d) 5
  17. 17.Which of these numbers is divisible by 11? (a) 1628 (b) 1637 (c) 1643 (d) 1627
  18. 18.Which of these numbers is divisible by 6? (a) 5351 (b) 5356 (c) 5343 (d) 5346
  19. 19.What is the smallest number you must add to 558 to make it divisible by 7? (a) 5 (b) 2 (c) 3 (d) 1
  20. 20.Which of these numbers is divisible by 8? (a) 4100 (b) 4083 (c) 4090 (d) 4088
  21. 21.What is the units digit of 9^10? (a) 9 (b) 2 (c) 1 (d) 0
  22. 22.What is the remainder when 727 is divided by 9? (a) 6 (b) 8 (c) 7 (d) 2
  23. 23.What is the remainder when 322 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
  24. 24.What is the units digit of 9^16? (a) 2 (b) 0 (c) 1 (d) 9
  25. 25.What is the remainder when 176 is divided by 3? (a) 0 (b) 1 (c) 2
  26. 26.What is the units digit of 7^32? (a) 7 (b) 1 (c) 9 (d) 3
  27. 27.What is the smallest number you must add to 772 to make it divisible by 10? (a) 7 (b) 9 (c) 8 (d) 2
  28. 28.Which of these numbers is divisible by 4? (a) 2241 (b) 2251 (c) 2247 (d) 2244
  29. 29.Which of these numbers is divisible by 11? (a) 2574 (b) 2581 (c) 2565 (d) 2592
  30. 30.What is the remainder when 152 is divided by 9? (a) 6 (b) 1 (c) 7 (d) 8
  31. 31.What is the smallest number you must add to 133 to make it divisible by 9? (a) 3 (b) 1 (c) 7 (d) 2
  32. 32.What is the smallest number you must add to 318 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
  33. 33.Which of these numbers is divisible by 8? (a) 6562 (b) 6554 (c) 6570 (d) 6560
  34. 34.Which of these numbers is divisible by 4? (a) 1404 (b) 1403 (c) 1407 (d) 1410
  35. 35.What is the smallest number you must add to 128 to make it divisible by 9? (a) 2 (b) 6 (c) 7 (d) 8
  36. 36.What is the remainder when 203 is divided by 3? (a) 2 (b) 0 (c) 1
  37. 37.Which of these numbers is divisible by 6? (a) 1528 (b) 1533 (c) 1524 (d) 1520
  38. 38.Which of these numbers is divisible by 9? (a) 7091 (b) 7074 (c) 7072 (d) 7076
  39. 39.Which of these numbers is divisible by 9? (a) 736 (b) 726 (c) 743 (d) 729
  40. 40.What is the remainder when 274 is divided by 8? (a) 2 (b) 3 (c) 6 (d) 1

Answer key

  1. 8
  2. 5797
  3. 1119
  4. 1396
  5. 2
  6. 1224
  7. 6
  8. 1
  9. 3
  10. 1
  11. 6
  12. 9
  13. 6705
  14. 1524
  15. 2
  16. 4
  17. 1628
  18. 5346
  19. 2
  20. 4088
  21. 1
  22. 7
  23. 2
  24. 1
  25. 2
  26. 1
  27. 8
  28. 2244
  29. 2574
  30. 8
  31. 2
  32. 2
  33. 6560
  34. 1404
  35. 7
  36. 2
  37. 1524
  38. 7074
  39. 729
  40. 2

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