Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 413 · 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.What is the smallest number you must add to 86 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
2. 2.What is the smallest number you must add to 212 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 6
3. 3.Which of these numbers is divisible by 9? (a) 722 (b) 717 (c) 720 (d) 735
4. 4.What is the smallest number you must add to 103 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
5. 5.What is the remainder when 156 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
6. 6.What is the remainder when 507 is divided by 9? (a) 6 (b) 2 (c) 3 (d) 4
7. 7.Which of these numbers is divisible by 11? (a) 5518 (b) 5500 (c) 5497 (d) 5506
8. 8.What is the units digit of 7^59? (a) 3 (b) 7 (c) 9 (d) 1
9. 9.What is the remainder when 394 is divided by 12? (a) 9 (b) 10 (c) 2 (d) 11
10. 10.Which of these numbers is divisible by 11? (a) 7427 (b) 7425 (c) 7437 (d) 7424
11. 11.What is the smallest number you must add to 204 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
12. 12.Which of these numbers is divisible by 3? (a) 1302 (b) 1306 (c) 1301 (d) 1304
13. 13.What is the remainder when 506 is divided by 8? (a) 1 (b) 2 (c) 3 (d) 6
14. 14.What is the smallest number you must add to 695 to make it divisible by 12? (a) 11 (b) 2 (c) 1 (d) 3
15. 15.What is the smallest number you must add to 551 to make it divisible by 8? (a) 2 (b) 7 (c) 1 (d) 3
16. 16.What is the smallest number you must add to 218 to make it divisible by 4? (a) 1 (b) 3 (c) 2
17. 17.What is the units digit of 7^21? (a) 3 (b) 1 (c) 7 (d) 9
18. 18.What is the remainder when 47 is divided by 3? (a) 0 (b) 2 (c) 1
19. 19.What is the units digit of 9^55? (a) 1 (b) 9 (c) 7 (d) 8
20. 20.What is the remainder when 151 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 2
21. 21.What is the smallest number you must add to 557 to make it divisible by 8? (a) 2 (b) 3 (c) 4 (d) 5
22. 22.What is the smallest number you must add to 100 to make it divisible by 3? (a) 2 (b) 1
23. 23.Which of these numbers is divisible by 3? (a) 1920 (b) 1919 (c) 1921 (d) 1924
24. 24.Which of these numbers is divisible by 6? (a) 3738 (b) 3748 (c) 3736 (d) 3739
25. 25.What is the units digit of 8^23? (a) 8 (b) 2 (c) 4 (d) 6
26. 26.What is the remainder when 465 is divided by 8? (a) 7 (b) 1 (c) 0 (d) 2
27. 27.What is the remainder when 321 is divided by 10? (a) 9 (b) 0 (c) 2 (d) 1
28. 28.What is the smallest number you must add to 213 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 2
29. 29.Which of these numbers is divisible by 9? (a) 5259 (b) 5240 (c) 5249 (d) 5247
30. 30.What is the remainder when 578 is divided by 12? (a) 2 (b) 3 (c) 10 (d) 1
31. 31.Which of these numbers is divisible by 4? (a) 1530 (b) 1533 (c) 1528 (d) 1527
32. 32.What is the remainder when 253 is divided by 7? (a) 2 (b) 6 (c) 1 (d) 0
33. 33.What is the units digit of 4^31? (a) 6 (b) 5 (c) 3 (d) 4
34. 34.What is the remainder when 187 is divided by 3? (a) 0 (b) 1 (c) 2
35. 35.What is the units digit of 7^43? (a) 9 (b) 7 (c) 3 (d) 1
36. 36.What is the remainder when 636 is divided by 8? (a) 5 (b) 4 (c) 3 (d) 6
37. 37.What is the units digit of 4^20? (a) 5 (b) 6 (c) 7 (d) 4
38. 38.Which of these numbers is divisible by 9? (a) 2967 (b) 2948 (c) 2952 (d) 2953
39. 39.Which of these numbers is divisible by 3? (a) 361 (b) 365 (c) 358 (d) 360
40. 40.What is the smallest number you must add to 310 to make it divisible by 4? (a) 3 (b) 2 (c) 1

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