Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 951 · 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 553 to make it divisible by 8? (a) 5 (b) 7 (c) 1 (d) 6
2. 2.What is the units digit of 9^57? (a) 8 (b) 7 (c) 9 (d) 1
3. 3.What is the units digit of 2^52? (a) 2 (b) 8 (c) 6 (d) 4
4. 4.What is the remainder when 176 is divided by 12? (a) 8 (b) 4 (c) 9 (d) 7
5. 5.What is the remainder when 56 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
6. 6.Which of these numbers is divisible by 11? (a) 2368 (b) 2376 (c) 2394 (d) 2386
7. 7.What is the units digit of 9^59? (a) 1 (b) 7 (c) 8 (d) 9
8. 8.What is the smallest number you must add to 289 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 4
9. 9.What is the units digit of 2^13? (a) 8 (b) 2 (c) 6 (d) 4
10. 10.What is the smallest number you must add to 404 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
11. 11.What is the units digit of 7^22? (a) 7 (b) 3 (c) 9 (d) 1
12. 12.What is the smallest number you must add to 732 to make it divisible by 10? (a) 7 (b) 8 (c) 9 (d) 2
13. 13.What is the smallest number you must add to 323 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 1
14. 14.Which of these numbers is divisible by 4? (a) 2592 (b) 2589 (c) 2597 (d) 2595
15. 15.What is the units digit of 7^38? (a) 1 (b) 9 (c) 3 (d) 7
16. 16.Which of these numbers is divisible by 8? (a) 2446 (b) 2450 (c) 2436 (d) 2440
17. 17.Which of these numbers is divisible by 9? (a) 1485 (b) 1490 (c) 1496 (d) 1481
18. 18.What is the smallest number you must add to 809 to make it divisible by 10? (a) 3 (b) 1 (c) 2 (d) 9
19. 19.Which of these numbers is divisible by 9? (a) 3957 (b) 3951 (c) 3948 (d) 3963
20. 20.What is the smallest number you must add to 309 to make it divisible by 8? (a) 2 (b) 4 (c) 5 (d) 3
21. 21.What is the units digit of 3^30? (a) 3 (b) 9 (c) 1 (d) 7
22. 22.What is the units digit of 8^21? (a) 2 (b) 4 (c) 6 (d) 8
23. 23.Which of these numbers is divisible by 4? (a) 858 (b) 852 (c) 853 (d) 849
24. 24.What is the smallest number you must add to 109 to make it divisible by 9? (a) 6 (b) 1 (c) 7 (d) 8
25. 25.Which of these numbers is divisible by 4? (a) 643 (b) 645 (c) 639 (d) 640
26. 26.What is the remainder when 694 is divided by 10? (a) 3 (b) 5 (c) 6 (d) 4
27. 27.Which of these numbers is divisible by 3? (a) 569 (b) 565 (c) 562 (d) 564
28. 28.Which of these numbers is divisible by 11? (a) 3166 (b) 3146 (c) 3136 (d) 3151
29. 29.What is the remainder when 613 is divided by 9? (a) 0 (b) 8 (c) 2 (d) 1
30. 30.Which of these numbers is divisible by 11? (a) 2272 (b) 2251 (c) 2258 (d) 2255
31. 31.What is the remainder when 125 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
32. 32.Which of these numbers is divisible by 6? (a) 1705 (b) 1697 (c) 1699 (d) 1698
33. 33.Which of these numbers is divisible by 9? (a) 5256 (b) 5250 (c) 5268 (d) 5258
34. 34.Which of these numbers is divisible by 6? (a) 4431 (b) 4428 (c) 4424 (d) 4437
35. 35.What is the units digit of 4^11? (a) 6 (b) 5 (c) 3 (d) 4
36. 36.What is the units digit of 3^58? (a) 1 (b) 9 (c) 7 (d) 3
37. 37.Which of these numbers is divisible by 11? (a) 1766 (b) 1773 (c) 1754 (d) 1760
38. 38.What is the smallest number you must add to 521 to make it divisible by 8? (a) 1 (b) 5 (c) 6 (d) 7
39. 39.What is the smallest number you must add to 404 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 6
40. 40.Which of these numbers is divisible by 9? (a) 6954 (b) 6962 (c) 6957 (d) 6971

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