Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 530 · 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 11? (a) 4136 (b) 4154 (c) 4133 (d) 4139
2. 2.What is the units digit of 3^16? (a) 9 (b) 3 (c) 1 (d) 7
3. 3.What is the remainder when 157 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
4. 4.Which of these numbers is divisible by 9? (a) 6694 (b) 6675 (c) 6686 (d) 6678
5. 5.What is the units digit of 2^41? (a) 6 (b) 8 (c) 4 (d) 2
6. 6.What is the units digit of 2^44? (a) 4 (b) 2 (c) 8 (d) 6
7. 7.What is the smallest number you must add to 110 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 5
8. 8.What is the smallest number you must add to 161 to make it divisible by 10? (a) 8 (b) 1 (c) 7 (d) 9
9. 9.What is the units digit of 4^25? (a) 6 (b) 5 (c) 3 (d) 4
10. 10.What is the units digit of 9^49? (a) 7 (b) 1 (c) 8 (d) 9
11. 11.What is the units digit of 3^38? (a) 1 (b) 3 (c) 7 (d) 9
12. 12.What is the remainder when 158 is divided by 12? (a) 1 (b) 3 (c) 10 (d) 2
13. 13.What is the units digit of 8^15? (a) 4 (b) 2 (c) 8 (d) 6
14. 14.What is the units digit of 7^26? (a) 9 (b) 7 (c) 1 (d) 3
15. 15.Which of these numbers is divisible by 3? (a) 538 (b) 541 (c) 540 (d) 544
16. 16.Which of these numbers is divisible by 9? (a) 1364 (b) 1354 (c) 1359 (d) 1374
17. 17.Which of these numbers is divisible by 3? (a) 494 (b) 496 (c) 492 (d) 491
18. 18.What is the remainder when 122 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
19. 19.Which of these numbers is divisible by 6? (a) 2675 (b) 2687 (c) 2676 (d) 2678
20. 20.What is the units digit of 3^35? (a) 9 (b) 7 (c) 1 (d) 3
21. 21.What is the units digit of 3^31? (a) 9 (b) 1 (c) 3 (d) 7
22. 22.What is the units digit of 2^17? (a) 8 (b) 6 (c) 2 (d) 4
23. 23.What is the units digit of 4^35? (a) 3 (b) 4 (c) 6 (d) 5
24. 24.What is the remainder when 299 is divided by 10? (a) 8 (b) 1 (c) 9 (d) 7
25. 25.What is the remainder when 551 is divided by 12? (a) 11 (b) 9 (c) 1 (d) 10
26. 26.What is the units digit of 8^59? (a) 4 (b) 2 (c) 8 (d) 6
27. 27.What is the units digit of 8^36? (a) 4 (b) 8 (c) 2 (d) 6
28. 28.What is the units digit of 2^33? (a) 6 (b) 2 (c) 4 (d) 8
29. 29.What is the units digit of 2^21? (a) 8 (b) 4 (c) 2 (d) 6
30. 30.What is the units digit of 7^51? (a) 3 (b) 7 (c) 1 (d) 9
31. 31.What is the units digit of 2^27? (a) 6 (b) 4 (c) 8 (d) 2
32. 32.What is the remainder when 188 is divided by 7? (a) 1 (b) 5 (c) 4 (d) 6
33. 33.Which of these numbers is divisible by 11? (a) 947 (b) 942 (c) 946 (d) 965
34. 34.What is the units digit of 4^60? (a) 5 (b) 6 (c) 7 (d) 4
35. 35.Which of these numbers is divisible by 6? (a) 635 (b) 630 (c) 638 (d) 625
36. 36.Which of these numbers is divisible by 8? (a) 2224 (b) 2236 (c) 2221 (d) 2229
37. 37.What is the smallest number you must add to 875 to make it divisible by 11? (a) 7 (b) 4 (c) 6 (d) 5
38. 38.What is the remainder when 38 is divided by 3? (a) 2 (b) 0 (c) 1
39. 39.Which of these numbers is divisible by 9? (a) 5473 (b) 5482 (c) 5465 (d) 5472
40. 40.What is the smallest number you must add to 460 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3

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