← Divisibility, remainders and units digits lesson New set →

Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 514 · 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 6? (a) 3029 (b) 3034 (c) 3039 (d) 3030
  2. 2.What is the smallest number you must add to 359 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 6
  3. 3.Which of these numbers is divisible by 6? (a) 5266 (b) 5268 (c) 5271 (d) 5275
  4. 4.What is the smallest number you must add to 444 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
  5. 5.What is the smallest number you must add to 459 to make it divisible by 7? (a) 3 (b) 5 (c) 4 (d) 2
  6. 6.Which of these numbers is divisible by 9? (a) 6678 (b) 6675 (c) 6693 (d) 6684
  7. 7.What is the smallest number you must add to 304 to make it divisible by 6? (a) 2 (b) 3 (c) 1 (d) 4
  8. 8.What is the remainder when 602 is divided by 8? (a) 1 (b) 3 (c) 6 (d) 2
  9. 9.What is the smallest number you must add to 82 to make it divisible by 8? (a) 5 (b) 7 (c) 6 (d) 2
  10. 10.What is the smallest number you must add to 398 to make it divisible by 8? (a) 2 (b) 6 (c) 3 (d) 1
  11. 11.What is the units digit of 9^50? (a) 1 (b) 2 (c) 0 (d) 9
  12. 12.What is the remainder when 193 is divided by 10? (a) 4 (b) 3 (c) 7 (d) 2
  13. 13.What is the remainder when 304 is divided by 9? (a) 2 (b) 6 (c) 7 (d) 8
  14. 14.What is the units digit of 7^40? (a) 3 (b) 9 (c) 7 (d) 1
  15. 15.What is the units digit of 7^59? (a) 3 (b) 9 (c) 7 (d) 1
  16. 16.What is the smallest number you must add to 158 to make it divisible by 3? (a) 2 (b) 1
  17. 17.Which of these numbers is divisible by 11? (a) 869 (b) 868 (c) 883 (d) 877
  18. 18.Which of these numbers is divisible by 3? (a) 2355 (b) 2356 (c) 2359 (d) 2353
  19. 19.Which of these numbers is divisible by 11? (a) 1887 (b) 1871 (c) 1869 (d) 1870
  20. 20.What is the smallest number you must add to 167 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
  21. 21.What is the smallest number you must add to 556 to make it divisible by 9? (a) 2 (b) 1 (c) 3 (d) 7
  22. 22.What is the units digit of 3^43? (a) 9 (b) 7 (c) 3 (d) 1
  23. 23.Which of these numbers is divisible by 11? (a) 9067 (b) 9045 (c) 9053 (d) 9062
  24. 24.What is the remainder when 350 is divided by 8? (a) 5 (b) 6 (c) 2 (d) 7
  25. 25.What is the smallest number you must add to 247 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
  26. 26.What is the smallest number you must add to 32 to make it divisible by 3? (a) 2 (b) 1
  27. 27.What is the smallest number you must add to 398 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
  28. 28.What is the units digit of 4^33? (a) 5 (b) 3 (c) 4 (d) 6
  29. 29.Which of these numbers is divisible by 6? (a) 5323 (b) 5311 (c) 5320 (d) 5316
  30. 30.What is the remainder when 107 is divided by 3? (a) 0 (b) 2 (c) 1
  31. 31.Which of these numbers is divisible by 11? (a) 2059 (b) 2071 (c) 2086 (d) 2068
  32. 32.What is the remainder when 156 is divided by 11? (a) 2 (b) 3 (c) 1 (d) 9
  33. 33.Which of these numbers is divisible by 3? (a) 2419 (b) 2417 (c) 2422 (d) 2418
  34. 34.Which of these numbers is divisible by 8? (a) 3202 (b) 3218 (c) 3209 (d) 3208
  35. 35.Which of these numbers is divisible by 11? (a) 5040 (b) 5028 (c) 5038 (d) 5052
  36. 36.What is the smallest number you must add to 823 to make it divisible by 12? (a) 4 (b) 5 (c) 6 (d) 7
  37. 37.Which of these numbers is divisible by 3? (a) 553 (b) 556 (c) 555 (d) 559
  38. 38.What is the units digit of 3^58? (a) 7 (b) 3 (c) 9 (d) 1
  39. 39.What is the remainder when 653 is divided by 11? (a) 7 (b) 3 (c) 4 (d) 5
  40. 40.What is the units digit of 4^50? (a) 7 (b) 4 (c) 5 (d) 6

Answer key

  1. 3030
  2. 5
  3. 5268
  4. 6
  5. 3
  6. 6678
  7. 2
  8. 2
  9. 6
  10. 2
  11. 1
  12. 3
  13. 7
  14. 1
  15. 3
  16. 1
  17. 869
  18. 2355
  19. 1870
  20. 3
  21. 2
  22. 7
  23. 9053
  24. 6
  25. 3
  26. 1
  27. 2
  28. 4
  29. 5316
  30. 2
  31. 2068
  32. 2
  33. 2418
  34. 3208
  35. 5038
  36. 5
  37. 555
  38. 9
  39. 4
  40. 6

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