Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 205 · 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 remainder when 137 is divided by 3? (a) 1 (b) 0 (c) 2
2. 2.What is the remainder when 181 is divided by 3? (a) 2 (b) 1 (c) 0
3. 3.Which of these numbers is divisible by 3? (a) 2686 (b) 2688 (c) 2689 (d) 2692
4. 4.What is the remainder when 400 is divided by 7? (a) 2 (b) 1 (c) 0 (d) 6
5. 5.What is the remainder when 594 is divided by 10? (a) 3 (b) 6 (c) 4 (d) 5
6. 6.What is the smallest number you must add to 251 to make it divisible by 11? (a) 1 (b) 2 (c) 9 (d) 3
7. 7.Which of these numbers is divisible by 3? (a) 2276 (b) 2279 (c) 2277 (d) 2282
8. 8.Which of these numbers is divisible by 9? (a) 819 (b) 815 (c) 821 (d) 829
9. 9.What is the remainder when 701 is divided by 12? (a) 6 (b) 7 (c) 5 (d) 4
10. 10.What is the smallest number you must add to 213 to make it divisible by 12? (a) 4 (b) 3 (c) 9 (d) 2
11. 11.Which of these numbers is divisible by 3? (a) 899 (b) 896 (c) 894 (d) 893
12. 12.Which of these numbers is divisible by 6? (a) 2279 (b) 2280 (c) 2289 (d) 2282
13. 13.What is the remainder when 213 is divided by 9? (a) 3 (b) 7 (c) 6 (d) 5
14. 14.What is the smallest number you must add to 470 to make it divisible by 7? (a) 1 (b) 6 (c) 5 (d) 4
15. 15.What is the units digit of 2^43? (a) 4 (b) 6 (c) 2 (d) 8
16. 16.What is the smallest number you must add to 439 to make it divisible by 10? (a) 9 (b) 3 (c) 1 (d) 2
17. 17.Which of these numbers is divisible by 3? (a) 914 (b) 916 (c) 912 (d) 911
18. 18.Which of these numbers is divisible by 4? (a) 378 (b) 373 (c) 372 (d) 370
19. 19.What is the units digit of 8^44? (a) 6 (b) 8 (c) 4 (d) 2
20. 20.What is the smallest number you must add to 287 to make it divisible by 4? (a) 2 (b) 3 (c) 1
21. 21.What is the remainder when 476 is divided by 6? (a) 1 (b) 4 (c) 2 (d) 3
22. 22.What is the units digit of 4^10? (a) 7 (b) 5 (c) 6 (d) 4
23. 23.What is the remainder when 776 is divided by 11? (a) 7 (b) 6 (c) 5 (d) 8
24. 24.What is the smallest number you must add to 211 to make it divisible by 10? (a) 1 (b) 8 (c) 7 (d) 9
25. 25.What is the units digit of 7^28? (a) 3 (b) 7 (c) 1 (d) 9
26. 26.Which of these numbers is divisible by 3? (a) 2157 (b) 2162 (c) 2155 (d) 2159
27. 27.What is the remainder when 321 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
28. 28.What is the smallest number you must add to 703 to make it divisible by 12? (a) 4 (b) 7 (c) 6 (d) 5
29. 29.Which of these numbers is divisible by 3? (a) 2662 (b) 2657 (c) 2659 (d) 2658
30. 30.What is the units digit of 8^51? (a) 8 (b) 2 (c) 6 (d) 4
31. 31.What is the remainder when 344 is divided by 9? (a) 7 (b) 3 (c) 1 (d) 2
32. 32.What is the units digit of 2^17? (a) 6 (b) 8 (c) 4 (d) 2
33. 33.What is the units digit of 8^55? (a) 8 (b) 4 (c) 2 (d) 6
34. 34.Which of these numbers is divisible by 3? (a) 530 (b) 526 (c) 523 (d) 525
35. 35.What is the units digit of 9^43? (a) 8 (b) 1 (c) 9 (d) 7
36. 36.What is the smallest number you must add to 185 to make it divisible by 10? (a) 7 (b) 6 (c) 4 (d) 5
37. 37.What is the remainder when 762 is divided by 12? (a) 5 (b) 7 (c) 6 (d) 8
38. 38.What is the remainder when 91 is divided by 3? (a) 2 (b) 1 (c) 0
39. 39.What is the remainder when 322 is divided by 11? (a) 4 (b) 8 (c) 3 (d) 2
40. 40.What is the smallest number you must add to 314 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3

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