Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 528 · 40 questions

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.

1. 1.Which of these numbers is divisible by 6? (a) 3302 (b) 3316 (c) 3310 (d) 3306
2. 2.What is the units digit of 3^30? (a) 7 (b) 9 (c) 1 (d) 3
3. 3.Which of these numbers is divisible by 11? (a) 7560 (b) 7553 (c) 7557 (d) 7575
4. 4.What is the smallest number you must add to 772 to make it divisible by 11? (a) 2 (b) 8 (c) 9 (d) 10
5. 5.What is the smallest number you must add to 566 to make it divisible by 10? (a) 6 (b) 5 (c) 3 (d) 4
6. 6.What is the units digit of 9^23? (a) 7 (b) 9 (c) 8 (d) 1
7. 7.Which of these numbers is divisible by 3? (a) 814 (b) 818 (c) 811 (d) 813
8. 8.What is the smallest number you must add to 460 to make it divisible by 9? (a) 7 (b) 6 (c) 8 (d) 1
9. 9.Which of these numbers is divisible by 9? (a) 6270 (b) 6262 (c) 6275 (d) 6264
10. 10.Which of these numbers is divisible by 8? (a) 4883 (b) 4872 (c) 4867 (d) 4875
11. 11.What is the units digit of 4^47? (a) 4 (b) 5 (c) 6 (d) 3
12. 12.Which of these numbers is divisible by 3? (a) 1732 (b) 1731 (c) 1735 (d) 1729
13. 13.Which of these numbers is divisible by 3? (a) 2261 (b) 2254 (c) 2256 (d) 2257
14. 14.Which of these numbers is divisible by 8? (a) 5791 (b) 5770 (c) 5776 (d) 5782
15. 15.What is the smallest number you must add to 244 to make it divisible by 6? (a) 4 (b) 1 (c) 2 (d) 3
16. 16.What is the remainder when 283 is divided by 6? (a) 1 (b) 0 (c) 5 (d) 2
17. 17.Which of these numbers is divisible by 4? (a) 1053 (b) 1047 (c) 1048 (d) 1051
18. 18.Which of these numbers is divisible by 4? (a) 1043 (b) 1044 (c) 1050 (d) 1045
19. 19.What is the smallest number you must add to 281 to make it divisible by 10? (a) 1 (b) 8 (c) 9 (d) 7
20. 20.What is the smallest number you must add to 543 to make it divisible by 10? (a) 3 (b) 7 (c) 8 (d) 6
21. 21.What is the remainder when 116 is divided by 3? (a) 0 (b) 1 (c) 2
22. 22.What is the smallest number you must add to 179 to make it divisible by 3? (a) 1 (b) 2
23. 23.What is the smallest number you must add to 124 to make it divisible by 12? (a) 9 (b) 4 (c) 7 (d) 8
24. 24.What is the smallest number you must add to 275 to make it divisible by 4? (a) 1 (b) 2 (c) 3
25. 25.What is the units digit of 7^46? (a) 7 (b) 1 (c) 9 (d) 3
26. 26.What is the remainder when 284 is divided by 11? (a) 8 (b) 2 (c) 9 (d) 10
27. 27.What is the units digit of 4^20? (a) 5 (b) 7 (c) 6 (d) 4
28. 28.Which of these numbers is divisible by 11? (a) 9207 (b) 9210 (c) 9224 (d) 9197
29. 29.What is the remainder when 439 is divided by 12? (a) 6 (b) 5 (c) 7 (d) 8
30. 30.What is the remainder when 451 is divided by 6? (a) 2 (b) 5 (c) 1 (d) 0
31. 31.What is the remainder when 836 is divided by 12? (a) 9 (b) 7 (c) 4 (d) 8
32. 32.What is the smallest number you must add to 227 to make it divisible by 4? (a) 3 (b) 2 (c) 1
33. 33.What is the units digit of 9^17? (a) 1 (b) 9 (c) 7 (d) 8
34. 34.What is the smallest number you must add to 693 to make it divisible by 12? (a) 3 (b) 4 (c) 2 (d) 9
35. 35.What is the units digit of 9^22? (a) 2 (b) 9 (c) 0 (d) 1
36. 36.What is the units digit of 2^51? (a) 4 (b) 2 (c) 6 (d) 8
37. 37.Which of these numbers is divisible by 6? (a) 5262 (b) 5260 (c) 5269 (d) 5266
38. 38.What is the units digit of 2^25? (a) 2 (b) 6 (c) 4 (d) 8
39. 39.What is the units digit of 8^27? (a) 2 (b) 4 (c) 8 (d) 6
40. 40.What is the smallest number you must add to 173 to make it divisible by 10? (a) 7 (b) 3 (c) 8 (d) 6

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