Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 212 · 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 9? (a) 7542 (b) 7540 (c) 7556 (d) 7543
2. 2.What is the smallest number you must add to 105 to make it divisible by 4? (a) 1 (b) 2 (c) 3
3. 3.What is the remainder when 226 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
4. 4.What is the units digit of 7^30? (a) 7 (b) 9 (c) 3 (d) 1
5. 5.Which of these numbers is divisible by 8? (a) 3229 (b) 3218 (c) 3216 (d) 3211
6. 6.Which of these numbers is divisible by 6? (a) 4663 (b) 4656 (c) 4653 (d) 4660
7. 7.Which of these numbers is divisible by 11? (a) 3845 (b) 3859 (c) 3865 (d) 3850
8. 8.What is the remainder when 548 is divided by 7? (a) 5 (b) 1 (c) 3 (d) 2
9. 9.What is the units digit of 3^55? (a) 1 (b) 7 (c) 9 (d) 3
10. 10.Which of these numbers is divisible by 6? (a) 3880 (b) 3882 (c) 3885 (d) 3891
11. 11.What is the remainder when 299 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 2
12. 12.What is the smallest number you must add to 838 to make it divisible by 12? (a) 2 (b) 10 (c) 1 (d) 3
13. 13.What is the smallest number you must add to 82 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 5
14. 14.What is the units digit of 2^20? (a) 2 (b) 4 (c) 6 (d) 8
15. 15.What is the smallest number you must add to 586 to make it divisible by 12? (a) 10 (b) 1 (c) 2 (d) 3
16. 16.What is the smallest number you must add to 158 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
17. 17.What is the remainder when 291 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
18. 18.What is the remainder when 469 is divided by 10? (a) 8 (b) 1 (c) 9 (d) 7
19. 19.What is the smallest number you must add to 381 to make it divisible by 7? (a) 3 (b) 4 (c) 5 (d) 6
20. 20.What is the smallest number you must add to 166 to make it divisible by 9? (a) 4 (b) 5 (c) 6 (d) 7
21. 21.What is the units digit of 8^34? (a) 4 (b) 8 (c) 2 (d) 6
22. 22.What is the remainder when 172 is divided by 3? (a) 0 (b) 2 (c) 1
23. 23.What is the units digit of 2^52? (a) 4 (b) 8 (c) 2 (d) 6
24. 24.What is the remainder when 111 is divided by 10? (a) 0 (b) 1 (c) 2 (d) 9
25. 25.Which of these numbers is divisible by 11? (a) 8130 (b) 8116 (c) 8118 (d) 8120
26. 26.What is the units digit of 8^21? (a) 4 (b) 2 (c) 8 (d) 6
27. 27.What is the smallest number you must add to 267 to make it divisible by 4? (a) 1 (b) 2 (c) 3
28. 28.What is the units digit of 3^25? (a) 1 (b) 7 (c) 9 (d) 3
29. 29.What is the remainder when 322 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
30. 30.What is the remainder when 115 is divided by 6? (a) 0 (b) 2 (c) 5 (d) 1
31. 31.What is the smallest number you must add to 269 to make it divisible by 11? (a) 5 (b) 8 (c) 7 (d) 6
32. 32.What is the units digit of 9^32? (a) 2 (b) 0 (c) 9 (d) 1
33. 33.Which of these numbers is divisible by 8? (a) 720 (b) 724 (c) 715 (d) 734
34. 34.Which of these numbers is divisible by 8? (a) 3352 (b) 3361 (c) 3354 (d) 3348
35. 35.What is the remainder when 691 is divided by 11? (a) 10 (b) 8 (c) 9 (d) 2
36. 36.What is the units digit of 2^47? (a) 4 (b) 8 (c) 2 (d) 6
37. 37.What is the units digit of 9^45? (a) 1 (b) 7 (c) 9 (d) 8
38. 38.Which of these numbers is divisible by 4? (a) 3270 (b) 3278 (c) 3274 (d) 3272
39. 39.What is the smallest number you must add to 322 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
40. 40.What is the remainder when 126 is divided by 8? (a) 2 (b) 7 (c) 5 (d) 6

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