Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 938 · 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 266 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
2. 2.What is the smallest number you must add to 124 to make it divisible by 7? (a) 1 (b) 3 (c) 5 (d) 2
3. 3.What is the units digit of 9^50? (a) 2 (b) 1 (c) 0 (d) 9
4. 4.What is the units digit of 7^35? (a) 3 (b) 9 (c) 7 (d) 1
5. 5.Which of these numbers is divisible by 4? (a) 916 (b) 922 (c) 917 (d) 915
6. 6.What is the units digit of 8^50? (a) 4 (b) 8 (c) 2 (d) 6
7. 7.Which of these numbers is divisible by 4? (a) 495 (b) 488 (c) 490 (d) 485
8. 8.Which of these numbers is divisible by 4? (a) 803 (b) 795 (c) 796 (d) 797
9. 9.What is the units digit of 2^43? (a) 2 (b) 4 (c) 8 (d) 6
10. 10.What is the units digit of 8^16? (a) 6 (b) 8 (c) 2 (d) 4
11. 11.What is the smallest number you must add to 152 to make it divisible by 3? (a) 2 (b) 1
12. 12.What is the units digit of 9^42? (a) 2 (b) 1 (c) 9 (d) 0
13. 13.What is the remainder when 261 is divided by 12? (a) 10 (b) 9 (c) 3 (d) 8
14. 14.Which of these numbers is divisible by 11? (a) 4752 (b) 4748 (c) 4764 (d) 4760
15. 15.Which of these numbers is divisible by 11? (a) 4239 (b) 4255 (c) 4267 (d) 4246
16. 16.What is the smallest number you must add to 563 to make it divisible by 11? (a) 10 (b) 9 (c) 2 (d) 8
17. 17.Which of these numbers is divisible by 8? (a) 3296 (b) 3303 (c) 3307 (d) 3293
18. 18.What is the remainder when 77 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
19. 19.What is the smallest number you must add to 633 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 3
20. 20.Which of these numbers is divisible by 4? (a) 765 (b) 760 (c) 758 (d) 763
21. 21.What is the remainder when 103 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
22. 22.What is the units digit of 9^43? (a) 9 (b) 8 (c) 7 (d) 1
23. 23.What is the units digit of 2^57? (a) 4 (b) 2 (c) 6 (d) 8
24. 24.Which of these numbers is divisible by 3? (a) 1771 (b) 1769 (c) 1766 (d) 1767
25. 25.Which of these numbers is divisible by 11? (a) 4471 (b) 4464 (c) 4486 (d) 4466
26. 26.What is the units digit of 9^11? (a) 8 (b) 9 (c) 7 (d) 1
27. 27.What is the units digit of 7^52? (a) 3 (b) 9 (c) 1 (d) 7
28. 28.What is the remainder when 106 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
29. 29.What is the units digit of 4^11? (a) 6 (b) 5 (c) 4 (d) 3
30. 30.Which of these numbers is divisible by 3? (a) 773 (b) 774 (c) 776 (d) 779
31. 31.Which of these numbers is divisible by 9? (a) 4318 (b) 4326 (c) 4332 (d) 4320
32. 32.Which of these numbers is divisible by 4? (a) 2338 (b) 2334 (c) 2331 (d) 2332
33. 33.What is the remainder when 681 is divided by 12? (a) 10 (b) 8 (c) 9 (d) 3
34. 34.What is the smallest number you must add to 402 to make it divisible by 10? (a) 7 (b) 8 (c) 2 (d) 9
35. 35.What is the smallest number you must add to 566 to make it divisible by 10? (a) 4 (b) 6 (c) 3 (d) 5
36. 36.What is the smallest number you must add to 274 to make it divisible by 4? (a) 1 (b) 3 (c) 2
37. 37.What is the units digit of 2^39? (a) 6 (b) 8 (c) 2 (d) 4
38. 38.Which of these numbers is divisible by 8? (a) 3673 (b) 3687 (c) 3680 (d) 3695
39. 39.Which of these numbers is divisible by 4? (a) 2852 (b) 2855 (c) 2858 (d) 2849
40. 40.What is the units digit of 3^56? (a) 1 (b) 9 (c) 3 (d) 7

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