Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 478 · 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 2^18? (a) 4 (b) 2 (c) 8 (d) 6
2. 2.What is the units digit of 8^17? (a) 8 (b) 6 (c) 4 (d) 2
3. 3.What is the units digit of 9^46? (a) 2 (b) 1 (c) 0 (d) 9
4. 4.What is the remainder when 183 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
5. 5.What is the units digit of 4^13? (a) 3 (b) 4 (c) 5 (d) 6
6. 6.What is the remainder when 719 is divided by 10? (a) 7 (b) 9 (c) 1 (d) 8
7. 7.What is the remainder when 610 is divided by 11? (a) 5 (b) 6 (c) 7 (d) 4
8. 8.Which of these numbers is divisible by 3? (a) 338 (b) 331 (c) 333 (d) 334
9. 9.Which of these numbers is divisible by 11? (a) 1254 (b) 1267 (c) 1245 (d) 1261
10. 10.What is the smallest number you must add to 439 to make it divisible by 9? (a) 1 (b) 7 (c) 2 (d) 3
11. 11.Which of these numbers is divisible by 3? (a) 1429 (b) 1432 (c) 1436 (d) 1431
12. 12.Which of these numbers is divisible by 8? (a) 2976 (b) 2970 (c) 2982 (d) 2990
13. 13.Which of these numbers is divisible by 9? (a) 6957 (b) 6970 (c) 6956 (d) 6960
14. 14.What is the units digit of 3^13? (a) 1 (b) 7 (c) 9 (d) 3
15. 15.Which of these numbers is divisible by 11? (a) 9526 (b) 9536 (c) 9543 (d) 9522
16. 16.What is the smallest number you must add to 382 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
17. 17.What is the remainder when 321 is divided by 11? (a) 9 (b) 3 (c) 1 (d) 2
18. 18.Which of these numbers is divisible by 11? (a) 4407 (b) 4397 (c) 4400 (d) 4418
19. 19.What is the remainder when 280 is divided by 9? (a) 2 (b) 8 (c) 0 (d) 1
20. 20.Which of these numbers is divisible by 3? (a) 2438 (b) 2432 (c) 2433 (d) 2435
21. 21.Which of these numbers is divisible by 3? (a) 729 (b) 730 (c) 728 (d) 733
22. 22.What is the remainder when 430 is divided by 12? (a) 11 (b) 10 (c) 2 (d) 9
23. 23.What is the remainder when 817 is divided by 11? (a) 4 (b) 2 (c) 3 (d) 8
24. 24.What is the units digit of 3^35? (a) 1 (b) 3 (c) 7 (d) 9
25. 25.Which of these numbers is divisible by 4? (a) 3055 (b) 3052 (c) 3051 (d) 3057
26. 26.Which of these numbers is divisible by 4? (a) 3099 (b) 3100 (c) 3107 (d) 3101
27. 27.What is the units digit of 4^22? (a) 5 (b) 6 (c) 7 (d) 4
28. 28.Which of these numbers is divisible by 11? (a) 7439 (b) 7430 (c) 7425 (d) 7421
29. 29.What is the units digit of 2^25? (a) 6 (b) 8 (c) 4 (d) 2
30. 30.What is the units digit of 8^30? (a) 8 (b) 2 (c) 6 (d) 4
31. 31.What is the remainder when 236 is divided by 3? (a) 1 (b) 2 (c) 0
32. 32.What is the units digit of 8^56? (a) 6 (b) 8 (c) 2 (d) 4
33. 33.Which of these numbers is divisible by 8? (a) 3028 (b) 3020 (c) 3016 (d) 3014
34. 34.What is the units digit of 4^46? (a) 7 (b) 6 (c) 4 (d) 5
35. 35.Which of these numbers is divisible by 11? (a) 4545 (b) 4555 (c) 4543 (d) 4538
36. 36.What is the units digit of 3^51? (a) 1 (b) 7 (c) 9 (d) 3
37. 37.What is the units digit of 4^58? (a) 5 (b) 4 (c) 6 (d) 7
38. 38.Which of these numbers is divisible by 6? (a) 583 (b) 598 (c) 591 (d) 588
39. 39.What is the units digit of 3^31? (a) 1 (b) 7 (c) 3 (d) 9
40. 40.Which of these numbers is divisible by 4? (a) 881 (b) 875 (c) 876 (d) 877

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