Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 632 · 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 477 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
2. 2.Which of these numbers is divisible by 8? (a) 1741 (b) 1748 (c) 1744 (d) 1759
3. 3.What is the units digit of 7^22? (a) 9 (b) 1 (c) 7 (d) 3
4. 4.What is the units digit of 8^21? (a) 6 (b) 2 (c) 4 (d) 8
5. 5.Which of these numbers is divisible by 9? (a) 1566 (b) 1567 (c) 1581 (d) 1558
6. 6.Which of these numbers is divisible by 11? (a) 3693 (b) 3674 (c) 3673 (d) 3678
7. 7.What is the remainder when 106 is divided by 3? (a) 2 (b) 0 (c) 1
8. 8.What is the units digit of 4^10? (a) 6 (b) 5 (c) 7 (d) 4
9. 9.What is the remainder when 151 is divided by 10? (a) 0 (b) 9 (c) 1 (d) 2
10. 10.What is the smallest number you must add to 166 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the units digit of 8^30? (a) 2 (b) 8 (c) 4 (d) 6
12. 12.What is the remainder when 207 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
13. 13.Which of these numbers is divisible by 9? (a) 1943 (b) 1944 (c) 1955 (d) 1950
14. 14.What is the units digit of 4^21? (a) 6 (b) 5 (c) 4 (d) 3
15. 15.What is the remainder when 443 is divided by 7? (a) 1 (b) 5 (c) 3 (d) 2
16. 16.What is the smallest number you must add to 321 to make it divisible by 12? (a) 9 (b) 4 (c) 2 (d) 3
17. 17.What is the units digit of 3^34? (a) 7 (b) 1 (c) 9 (d) 3
18. 18.Which of these numbers is divisible by 4? (a) 406 (b) 408 (c) 409 (d) 414
19. 19.What is the remainder when 78 is divided by 7? (a) 6 (b) 1 (c) 2 (d) 0
20. 20.What is the remainder when 177 is divided by 8? (a) 0 (b) 7 (c) 2 (d) 1
21. 21.What is the remainder when 361 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
22. 22.Which of these numbers is divisible by 9? (a) 6397 (b) 6399 (c) 6409 (d) 6403
23. 23.What is the remainder when 391 is divided by 8? (a) 6 (b) 5 (c) 7 (d) 1
24. 24.What is the remainder when 349 is divided by 8? (a) 4 (b) 6 (c) 3 (d) 5
25. 25.Which of these numbers is divisible by 9? (a) 6569 (b) 6561 (c) 6576 (d) 6558
26. 26.What is the remainder when 635 is divided by 8? (a) 4 (b) 2 (c) 3 (d) 5
27. 27.Which of these numbers is divisible by 9? (a) 2504 (b) 2502 (c) 2497 (d) 2513
28. 28.Which of these numbers is divisible by 8? (a) 1032 (b) 1035 (c) 1031 (d) 1041
29. 29.What is the smallest number you must add to 148 to make it divisible by 12? (a) 9 (b) 7 (c) 4 (d) 8
30. 30.What is the smallest number you must add to 315 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
31. 31.Which of these numbers is divisible by 11? (a) 3143 (b) 3148 (c) 3167 (d) 3146
32. 32.What is the smallest number you must add to 122 to make it divisible by 12? (a) 9 (b) 11 (c) 10 (d) 2
33. 33.Which of these numbers is divisible by 4? (a) 365 (b) 375 (c) 368 (d) 369
34. 34.What is the remainder when 77 is divided by 3? (a) 1 (b) 0 (c) 2
35. 35.What is the remainder when 680 is divided by 9? (a) 6 (b) 7 (c) 4 (d) 5
36. 36.What is the units digit of 2^39? (a) 8 (b) 4 (c) 2 (d) 6
37. 37.What is the units digit of 7^54? (a) 3 (b) 9 (c) 1 (d) 7
38. 38.What is the units digit of 2^59? (a) 2 (b) 4 (c) 6 (d) 8
39. 39.Which of these numbers is divisible by 3? (a) 842 (b) 847 (c) 843 (d) 845
40. 40.Which of these numbers is divisible by 11? (a) 2723 (b) 2706 (c) 2704 (d) 2711

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