Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 53 · 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.Which of these numbers is divisible by 6? (a) 874 (b) 859 (c) 864 (d) 868
2. 2.What is the smallest number you must add to 480 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 3
3. 3.What is the units digit of 9^26? (a) 2 (b) 9 (c) 0 (d) 1
4. 4.What is the remainder when 502 is divided by 7? (a) 5 (b) 4 (c) 2 (d) 6
5. 5.What is the smallest number you must add to 65 to make it divisible by 3? (a) 2 (b) 1
6. 6.What is the smallest number you must add to 280 to make it divisible by 12? (a) 9 (b) 8 (c) 4 (d) 7
7. 7.Which of these numbers is divisible by 9? (a) 665 (b) 657 (c) 667 (d) 651
8. 8.Which of these numbers is divisible by 6? (a) 1775 (b) 1769 (c) 1781 (d) 1770
9. 9.What is the remainder when 202 is divided by 3? (a) 1 (b) 2 (c) 0
10. 10.Which of these numbers is divisible by 6? (a) 3836 (b) 3840 (c) 3847 (d) 3842
11. 11.What is the units digit of 8^52? (a) 8 (b) 6 (c) 4 (d) 2
12. 12.Which of these numbers is divisible by 6? (a) 4094 (b) 4101 (c) 4092 (d) 4091
13. 13.What is the remainder when 32 is divided by 3? (a) 0 (b) 2 (c) 1
14. 14.Which of these numbers is divisible by 11? (a) 8030 (b) 8029 (c) 8045 (d) 8035
15. 15.What is the smallest number you must add to 56 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
16. 16.What is the units digit of 2^58? (a) 6 (b) 8 (c) 2 (d) 4
17. 17.Which of these numbers is divisible by 8? (a) 6941 (b) 6944 (c) 6957 (d) 6948
18. 18.What is the units digit of 2^31? (a) 2 (b) 6 (c) 4 (d) 8
19. 19.Which of these numbers is divisible by 3? (a) 344 (b) 346 (c) 350 (d) 345
20. 20.What is the remainder when 445 is divided by 9? (a) 6 (b) 4 (c) 3 (d) 5
21. 21.What is the smallest number you must add to 286 to make it divisible by 8? (a) 2 (b) 6 (c) 1 (d) 3
22. 22.What is the remainder when 517 is divided by 7? (a) 4 (b) 1 (c) 6 (d) 5
23. 23.What is the smallest number you must add to 233 to make it divisible by 9? (a) 8 (b) 3 (c) 2 (d) 1
24. 24.What is the units digit of 9^45? (a) 1 (b) 8 (c) 9 (d) 7
25. 25.What is the units digit of 9^11? (a) 1 (b) 8 (c) 9 (d) 7
26. 26.What is the units digit of 8^43? (a) 6 (b) 4 (c) 2 (d) 8
27. 27.What is the remainder when 148 is divided by 10? (a) 2 (b) 7 (c) 9 (d) 8
28. 28.What is the remainder when 221 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
29. 29.What is the remainder when 116 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 3
30. 30.What is the remainder when 431 is divided by 11? (a) 9 (b) 2 (c) 1 (d) 3
31. 31.What is the units digit of 8^27? (a) 6 (b) 8 (c) 2 (d) 4
32. 32.What is the units digit of 3^54? (a) 3 (b) 1 (c) 9 (d) 7
33. 33.What is the remainder when 256 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
34. 34.What is the units digit of 4^36? (a) 5 (b) 4 (c) 6 (d) 7
35. 35.What is the units digit of 2^39? (a) 2 (b) 6 (c) 4 (d) 8
36. 36.What is the remainder when 185 is divided by 8? (a) 0 (b) 2 (c) 7 (d) 1
37. 37.Which of these numbers is divisible by 3? (a) 1684 (b) 1679 (c) 1680 (d) 1682
38. 38.Which of these numbers is divisible by 6? (a) 2490 (b) 2493 (c) 2485 (d) 2500
39. 39.What is the remainder when 530 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 2
40. 40.What is the smallest number you must add to 161 to make it divisible by 3? (a) 2 (b) 1

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