Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 439 · 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 569 is divided by 10? (a) 8 (b) 1 (c) 9 (d) 7
2. 2.What is the remainder when 262 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
3. 3.Which of these numbers is divisible by 4? (a) 510 (b) 519 (c) 513 (d) 512
4. 4.Which of these numbers is divisible by 4? (a) 3497 (b) 3491 (c) 3492 (d) 3495
5. 5.What is the units digit of 3^13? (a) 7 (b) 9 (c) 3 (d) 1
6. 6.Which of these numbers is divisible by 8? (a) 607 (b) 622 (c) 608 (d) 612
7. 7.Which of these numbers is divisible by 11? (a) 2342 (b) 2356 (c) 2343 (d) 2347
8. 8.Which of these numbers is divisible by 8? (a) 736 (b) 742 (c) 747 (d) 733
9. 9.What is the smallest number you must add to 271 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 1
10. 10.What is the remainder when 424 is divided by 9? (a) 2 (b) 1 (c) 0 (d) 8
11. 11.Which of these numbers is divisible by 6? (a) 1368 (b) 1371 (c) 1379 (d) 1365
12. 12.What is the smallest number you must add to 398 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
13. 13.What is the smallest number you must add to 705 to make it divisible by 11? (a) 1 (b) 10 (c) 8 (d) 9
14. 14.Which of these numbers is divisible by 11? (a) 2963 (b) 2947 (c) 2950 (d) 2948
15. 15.What is the remainder when 812 is divided by 11? (a) 10 (b) 8 (c) 2 (d) 9
16. 16.What is the smallest number you must add to 363 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
17. 17.What is the smallest number you must add to 194 to make it divisible by 9? (a) 4 (b) 5 (c) 6 (d) 3
18. 18.What is the remainder when 233 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
19. 19.What is the smallest number you must add to 99 to make it divisible by 4? (a) 1 (b) 3 (c) 2
20. 20.What is the remainder when 242 is divided by 3? (a) 1 (b) 2 (c) 0
21. 21.Which of these numbers is divisible by 9? (a) 5749 (b) 5755 (c) 5737 (d) 5742
22. 22.What is the remainder when 536 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 3
23. 23.What is the smallest number you must add to 762 to make it divisible by 10? (a) 7 (b) 9 (c) 8 (d) 2
24. 24.What is the units digit of 9^55? (a) 9 (b) 7 (c) 8 (d) 1
25. 25.What is the smallest number you must add to 463 to make it divisible by 8? (a) 1 (b) 2 (c) 3 (d) 7
26. 26.What is the remainder when 504 is divided by 11? (a) 8 (b) 2 (c) 9 (d) 10
27. 27.What is the units digit of 7^42? (a) 1 (b) 3 (c) 7 (d) 9
28. 28.Which of these numbers is divisible by 11? (a) 4303 (b) 4314 (c) 4312 (d) 4324
29. 29.What is the units digit of 4^60? (a) 6 (b) 7 (c) 4 (d) 5
30. 30.What is the remainder when 522 is divided by 7? (a) 6 (b) 5 (c) 3 (d) 4
31. 31.What is the remainder when 56 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
32. 32.Which of these numbers is divisible by 9? (a) 1232 (b) 1222 (c) 1208 (d) 1215
33. 33.What is the units digit of 9^13? (a) 9 (b) 8 (c) 1 (d) 7
34. 34.What is the remainder when 684 is divided by 11? (a) 9 (b) 2 (c) 3 (d) 1
35. 35.What is the smallest number you must add to 155 to make it divisible by 9? (a) 2 (b) 8 (c) 6 (d) 7
36. 36.Which of these numbers is divisible by 3? (a) 190 (b) 186 (c) 188 (d) 184
37. 37.What is the units digit of 8^50? (a) 6 (b) 8 (c) 2 (d) 4
38. 38.Which of these numbers is divisible by 4? (a) 2231 (b) 2225 (c) 2221 (d) 2224
39. 39.What is the remainder when 32 is divided by 3? (a) 0 (b) 2 (c) 1
40. 40.Which of these numbers is divisible by 6? (a) 3703 (b) 3700 (c) 3710 (d) 3702

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