Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 385 · 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 units digit of 8^32? (a) 2 (b) 4 (c) 6 (d) 8
2. 2.Which of these numbers is divisible by 4? (a) 1843 (b) 1833 (c) 1836 (d) 1839
3. 3.What is the remainder when 872 is divided by 12? (a) 9 (b) 8 (c) 7 (d) 4
4. 4.Which of these numbers is divisible by 9? (a) 2590 (b) 2603 (c) 2592 (d) 2597
5. 5.What is the units digit of 4^21? (a) 5 (b) 3 (c) 4 (d) 6
6. 6.Which of these numbers is divisible by 11? (a) 4282 (b) 4297 (c) 4277 (d) 4279
7. 7.What is the smallest number you must add to 186 to make it divisible by 10? (a) 4 (b) 6 (c) 5 (d) 3
8. 8.What is the remainder when 317 is divided by 11? (a) 10 (b) 2 (c) 8 (d) 9
9. 9.What is the remainder when 405 is divided by 10? (a) 5 (b) 7 (c) 6 (d) 4
10. 10.What is the remainder when 58 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
11. 11.What is the smallest number you must add to 793 to make it divisible by 11? (a) 9 (b) 1 (c) 8 (d) 10
12. 12.What is the units digit of 8^22? (a) 8 (b) 2 (c) 6 (d) 4
13. 13.Which of these numbers is divisible by 11? (a) 4642 (b) 4646 (c) 4655 (d) 4638
14. 14.What is the units digit of 4^20? (a) 4 (b) 6 (c) 7 (d) 5
15. 15.Which of these numbers is divisible by 8? (a) 3956 (b) 3970 (c) 3960 (d) 3967
16. 16.What is the smallest number you must add to 665 to make it divisible by 11? (a) 5 (b) 6 (c) 7 (d) 8
17. 17.What is the units digit of 4^45? (a) 4 (b) 5 (c) 6 (d) 3
18. 18.What is the units digit of 2^47? (a) 4 (b) 6 (c) 8 (d) 2
19. 19.What is the units digit of 2^20? (a) 2 (b) 4 (c) 8 (d) 6
20. 20.What is the smallest number you must add to 154 to make it divisible by 12? (a) 1 (b) 3 (c) 2 (d) 10
21. 21.Which of these numbers is divisible by 9? (a) 2278 (b) 2268 (c) 2266 (d) 2276
22. 22.Which of these numbers is divisible by 3? (a) 521 (b) 518 (c) 524 (d) 519
23. 23.What is the units digit of 9^37? (a) 1 (b) 7 (c) 8 (d) 9
24. 24.Which of these numbers is divisible by 3? (a) 1941 (b) 1942 (c) 1940 (d) 1945
25. 25.What is the remainder when 658 is divided by 10? (a) 9 (b) 8 (c) 2 (d) 7
26. 26.What is the remainder when 146 is divided by 3? (a) 1 (b) 0 (c) 2
27. 27.What is the units digit of 4^31? (a) 3 (b) 5 (c) 4 (d) 6
28. 28.What is the smallest number you must add to 192 to make it divisible by 10? (a) 2 (b) 9 (c) 7 (d) 8
29. 29.Which of these numbers is divisible by 11? (a) 8438 (b) 8429 (c) 8423 (d) 8426
30. 30.What is the units digit of 9^17? (a) 9 (b) 1 (c) 7 (d) 8
31. 31.What is the remainder when 224 is divided by 10? (a) 3 (b) 4 (c) 6 (d) 5
32. 32.What is the remainder when 188 is divided by 3? (a) 1 (b) 0 (c) 2
33. 33.What is the units digit of 4^58? (a) 4 (b) 6 (c) 7 (d) 5
34. 34.What is the units digit of 7^37? (a) 1 (b) 3 (c) 9 (d) 7
35. 35.What is the units digit of 2^18? (a) 8 (b) 6 (c) 2 (d) 4
36. 36.What is the units digit of 9^50? (a) 0 (b) 9 (c) 2 (d) 1
37. 37.Which of these numbers is divisible by 11? (a) 7792 (b) 7777 (c) 7773 (d) 7784
38. 38.What is the units digit of 3^53? (a) 3 (b) 9 (c) 1 (d) 7
39. 39.Which of these numbers is divisible by 3? (a) 1745 (b) 1751 (c) 1748 (d) 1746
40. 40.What is the units digit of 9^58? (a) 9 (b) 2 (c) 1 (d) 0

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