Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 673 · 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 7^13? (a) 9 (b) 3 (c) 7 (d) 1
2. 2.What is the remainder when 226 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
3. 3.What is the units digit of 9^60? (a) 0 (b) 1 (c) 9 (d) 2
4. 4.What is the units digit of 9^43? (a) 7 (b) 1 (c) 9 (d) 8
5. 5.Which of these numbers is divisible by 8? (a) 2573 (b) 2581 (c) 2568 (d) 2563
6. 6.What is the remainder when 558 is divided by 8? (a) 7 (b) 5 (c) 2 (d) 6
7. 7.What is the remainder when 515 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
8. 8.What is the remainder when 216 is divided by 11? (a) 8 (b) 4 (c) 6 (d) 7
9. 9.What is the smallest number you must add to 115 to make it divisible by 6? (a) 4 (b) 5 (c) 1 (d) 3
10. 10.What is the smallest number you must add to 103 to make it divisible by 9? (a) 5 (b) 6 (c) 7 (d) 4
11. 11.What is the units digit of 2^30? (a) 2 (b) 6 (c) 8 (d) 4
12. 12.What is the units digit of 8^12? (a) 6 (b) 2 (c) 8 (d) 4
13. 13.Which of these numbers is divisible by 9? (a) 2493 (b) 2505 (c) 2494 (d) 2491
14. 14.Which of these numbers is divisible by 11? (a) 1889 (b) 1873 (c) 1896 (d) 1881
15. 15.What is the units digit of 2^44? (a) 6 (b) 4 (c) 2 (d) 8
16. 16.What is the remainder when 258 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
17. 17.What is the remainder when 173 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
18. 18.Which of these numbers is divisible by 3? (a) 906 (b) 907 (c) 905 (d) 911
19. 19.What is the units digit of 2^53? (a) 4 (b) 2 (c) 8 (d) 6
20. 20.Which of these numbers is divisible by 8? (a) 1512 (b) 1516 (c) 1508 (d) 1524
21. 21.What is the remainder when 436 is divided by 11? (a) 4 (b) 6 (c) 7 (d) 8
22. 22.What is the units digit of 3^21? (a) 7 (b) 3 (c) 1 (d) 9
23. 23.What is the smallest number you must add to 239 to make it divisible by 9? (a) 6 (b) 5 (c) 4 (d) 3
24. 24.Which of these numbers is divisible by 6? (a) 5303 (b) 5298 (c) 5308 (d) 5297
25. 25.Which of these numbers is divisible by 4? (a) 431 (b) 427 (c) 421 (d) 424
26. 26.What is the remainder when 243 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
27. 27.What is the smallest number you must add to 164 to make it divisible by 3? (a) 2 (b) 1
28. 28.What is the smallest number you must add to 616 to make it divisible by 10? (a) 3 (b) 5 (c) 6 (d) 4
29. 29.What is the remainder when 214 is divided by 8? (a) 5 (b) 2 (c) 7 (d) 6
30. 30.What is the smallest number you must add to 555 to make it divisible by 10? (a) 4 (b) 5 (c) 6 (d) 7
31. 31.What is the smallest number you must add to 443 to make it divisible by 12? (a) 3 (b) 11 (c) 2 (d) 1
32. 32.Which of these numbers is divisible by 8? (a) 3202 (b) 3218 (c) 3211 (d) 3208
33. 33.Which of these numbers is divisible by 4? (a) 2559 (b) 2551 (c) 2553 (d) 2552
34. 34.What is the remainder when 41 is divided by 3? (a) 0 (b) 2 (c) 1
35. 35.What is the remainder when 551 is divided by 8? (a) 5 (b) 7 (c) 1 (d) 6
36. 36.What is the smallest number you must add to 75 to make it divisible by 7? (a) 5 (b) 3 (c) 1 (d) 2
37. 37.What is the units digit of 9^21? (a) 8 (b) 1 (c) 7 (d) 9
38. 38.What is the remainder when 727 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 2
39. 39.Which of these numbers is divisible by 3? (a) 228 (b) 233 (c) 230 (d) 227
40. 40.Which of these numbers is divisible by 6? (a) 1699 (b) 1693 (c) 1692 (d) 1687

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