Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 7 · 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 275 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 1
2. 2.Which of these numbers is divisible by 3? (a) 648 (b) 649 (c) 652 (d) 647
3. 3.Which of these numbers is divisible by 6? (a) 3350 (b) 3344 (c) 3348 (d) 3357
4. 4.What is the remainder when 236 is divided by 10? (a) 5 (b) 7 (c) 6 (d) 4
5. 5.What is the units digit of 3^33? (a) 3 (b) 9 (c) 1 (d) 7
6. 6.What is the units digit of 8^32? (a) 4 (b) 2 (c) 8 (d) 6
7. 7.Which of these numbers is divisible by 8? (a) 7029 (b) 7016 (c) 7021 (d) 7009
8. 8.What is the remainder when 555 is divided by 12? (a) 2 (b) 9 (c) 4 (d) 3
9. 9.What is the remainder when 606 is divided by 11? (a) 10 (b) 1 (c) 0 (d) 2
10. 10.What is the remainder when 161 is divided by 12? (a) 5 (b) 6 (c) 4 (d) 7
11. 11.What is the units digit of 2^60? (a) 8 (b) 4 (c) 6 (d) 2
12. 12.What is the units digit of 3^12? (a) 9 (b) 7 (c) 3 (d) 1
13. 13.What is the remainder when 212 is divided by 7? (a) 5 (b) 1 (c) 3 (d) 2
14. 14.What is the smallest number you must add to 350 to make it divisible by 11? (a) 9 (b) 2 (c) 3 (d) 1
15. 15.Which of these numbers is divisible by 3? (a) 2362 (b) 2361 (c) 2360 (d) 2366
16. 16.What is the units digit of 7^34? (a) 9 (b) 7 (c) 1 (d) 3
17. 17.What is the smallest number you must add to 261 to make it divisible by 7? (a) 6 (b) 2 (c) 4 (d) 5
18. 18.What is the units digit of 9^57? (a) 8 (b) 7 (c) 1 (d) 9
19. 19.Which of these numbers is divisible by 8? (a) 2714 (b) 2699 (c) 2707 (d) 2704
20. 20.Which of these numbers is divisible by 11? (a) 3169 (b) 3166 (c) 3183 (d) 3168
21. 21.What is the units digit of 3^17? (a) 9 (b) 7 (c) 1 (d) 3
22. 22.What is the remainder when 205 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
23. 23.What is the units digit of 4^12? (a) 4 (b) 6 (c) 5 (d) 7
24. 24.Which of these numbers is divisible by 6? (a) 2915 (b) 2921 (c) 2926 (d) 2916
25. 25.What is the units digit of 4^27? (a) 5 (b) 6 (c) 3 (d) 4
26. 26.What is the units digit of 3^53? (a) 3 (b) 9 (c) 7 (d) 1
27. 27.What is the units digit of 7^35? (a) 7 (b) 1 (c) 3 (d) 9
28. 28.Which of these numbers is divisible by 11? (a) 8609 (b) 8621 (c) 8602 (d) 8594
29. 29.What is the remainder when 81 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
30. 30.Which of these numbers is divisible by 8? (a) 5483 (b) 5497 (c) 5489 (d) 5488
31. 31.What is the remainder when 768 is divided by 10? (a) 9 (b) 8 (c) 7 (d) 2
32. 32.Which of these numbers is divisible by 8? (a) 6805 (b) 6815 (c) 6800 (d) 6798
33. 33.What is the smallest number you must add to 475 to make it divisible by 7? (a) 1 (b) 3 (c) 6 (d) 2
34. 34.Which of these numbers is divisible by 8? (a) 3264 (b) 3274 (c) 3269 (d) 3263
35. 35.What is the remainder when 390 is divided by 9? (a) 2 (b) 6 (c) 3 (d) 4
36. 36.What is the remainder when 156 is divided by 7? (a) 5 (b) 3 (c) 1 (d) 2
37. 37.Which of these numbers is divisible by 4? (a) 796 (b) 801 (c) 795 (d) 797
38. 38.What is the smallest number you must add to 292 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
39. 39.What is the units digit of 9^27? (a) 7 (b) 1 (c) 8 (d) 9
40. 40.Which of these numbers is divisible by 8? (a) 1668 (b) 1659 (c) 1656 (d) 1655

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