Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 793 · 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 3^60? (a) 7 (b) 9 (c) 1 (d) 3
2. 2.Which of these numbers is divisible by 3? (a) 1536 (b) 1534 (c) 1538 (d) 1541
3. 3.What is the smallest number you must add to 144 to make it divisible by 10? (a) 7 (b) 5 (c) 4 (d) 6
4. 4.What is the smallest number you must add to 227 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 1
5. 5.What is the units digit of 9^15? (a) 1 (b) 9 (c) 8 (d) 7
6. 6.Which of these numbers is divisible by 11? (a) 3086 (b) 3092 (c) 3080 (d) 3078
7. 7.What is the smallest number you must add to 277 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 1
8. 8.Which of these numbers is divisible by 9? (a) 6321 (b) 6315 (c) 6318 (d) 6333
9. 9.What is the remainder when 153 is divided by 10? (a) 2 (b) 3 (c) 4 (d) 7
10. 10.What is the smallest number you must add to 303 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 2
11. 11.What is the units digit of 2^20? (a) 4 (b) 2 (c) 8 (d) 6
12. 12.What is the smallest number you must add to 650 to make it divisible by 12? (a) 11 (b) 9 (c) 10 (d) 2
13. 13.Which of these numbers is divisible by 4? (a) 2647 (b) 2648 (c) 2653 (d) 2650
14. 14.What is the smallest number you must add to 311 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
15. 15.Which of these numbers is divisible by 4? (a) 1278 (b) 1281 (c) 1274 (d) 1276
16. 16.Which of these numbers is divisible by 8? (a) 6372 (b) 6360 (c) 6357 (d) 6363
17. 17.What is the units digit of 8^11? (a) 4 (b) 8 (c) 6 (d) 2
18. 18.What is the units digit of 8^28? (a) 6 (b) 8 (c) 2 (d) 4
19. 19.What is the remainder when 105 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
20. 20.What is the units digit of 2^39? (a) 4 (b) 8 (c) 6 (d) 2
21. 21.What is the smallest number you must add to 613 to make it divisible by 10? (a) 3 (b) 8 (c) 6 (d) 7
22. 22.Which of these numbers is divisible by 9? (a) 4189 (b) 4202 (c) 4178 (d) 4185
23. 23.Which of these numbers is divisible by 11? (a) 958 (b) 957 (c) 954 (d) 969
24. 24.What is the units digit of 4^10? (a) 5 (b) 7 (c) 4 (d) 6
25. 25.What is the remainder when 122 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
26. 26.What is the units digit of 8^35? (a) 6 (b) 4 (c) 2 (d) 8
27. 27.What is the units digit of 3^52? (a) 7 (b) 1 (c) 9 (d) 3
28. 28.What is the units digit of 4^35? (a) 3 (b) 4 (c) 5 (d) 6
29. 29.What is the units digit of 9^25? (a) 8 (b) 1 (c) 9 (d) 7
30. 30.Which of these numbers is divisible by 8? (a) 3808 (b) 3814 (c) 3803 (d) 3818
31. 31.Which of these numbers is divisible by 9? (a) 7058 (b) 7081 (c) 7073 (d) 7065
32. 32.What is the remainder when 313 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
33. 33.What is the units digit of 7^44? (a) 7 (b) 9 (c) 3 (d) 1
34. 34.Which of these numbers is divisible by 3? (a) 1334 (b) 1337 (c) 1340 (d) 1335
35. 35.What is the smallest number you must add to 321 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
36. 36.What is the remainder when 139 is divided by 12? (a) 8 (b) 6 (c) 7 (d) 5
37. 37.Which of these numbers is divisible by 6? (a) 1087 (b) 1076 (c) 1085 (d) 1080
38. 38.Which of these numbers is divisible by 3? (a) 701 (b) 702 (c) 704 (d) 706
39. 39.What is the units digit of 8^56? (a) 4 (b) 2 (c) 8 (d) 6
40. 40.What is the units digit of 7^22? (a) 1 (b) 9 (c) 3 (d) 7

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