Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 566 · 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 6? (a) 2352 (b) 2360 (c) 2348 (d) 2355
2. 2.Which of these numbers is divisible by 9? (a) 7540 (b) 7525 (c) 7517 (d) 7524
3. 3.What is the remainder when 397 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 6
4. 4.What is the remainder when 107 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
5. 5.What is the units digit of 7^55? (a) 9 (b) 7 (c) 3 (d) 1
6. 6.Which of these numbers is divisible by 9? (a) 5167 (b) 5158 (c) 5166 (d) 5178
7. 7.What is the remainder when 573 is divided by 11? (a) 1 (b) 0 (c) 10 (d) 2
8. 8.What is the smallest number you must add to 148 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 1
9. 9.What is the units digit of 4^56? (a) 4 (b) 6 (c) 5 (d) 7
10. 10.What is the units digit of 8^37? (a) 2 (b) 8 (c) 6 (d) 4
11. 11.Which of these numbers is divisible by 8? (a) 3376 (b) 3381 (c) 3389 (d) 3372
12. 12.What is the remainder when 606 is divided by 8? (a) 2 (b) 7 (c) 6 (d) 5
13. 13.What is the units digit of 2^30? (a) 6 (b) 2 (c) 4 (d) 8
14. 14.Which of these numbers is divisible by 11? (a) 8928 (b) 8935 (c) 8921 (d) 8911
15. 15.Which of these numbers is divisible by 3? (a) 1004 (b) 1006 (c) 1002 (d) 1001
16. 16.Which of these numbers is divisible by 11? (a) 2787 (b) 2763 (c) 2772 (d) 2780
17. 17.What is the smallest number you must add to 115 to make it divisible by 11? (a) 8 (b) 6 (c) 5 (d) 7
18. 18.Which of these numbers is divisible by 8? (a) 800 (b) 795 (c) 802 (d) 814
19. 19.Which of these numbers is divisible by 9? (a) 2220 (b) 2205 (c) 2201 (d) 2212
20. 20.What is the smallest number you must add to 707 to make it divisible by 12? (a) 3 (b) 2 (c) 11 (d) 1
21. 21.What is the remainder when 141 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
22. 22.Which of these numbers is divisible by 4? (a) 176 (b) 177 (c) 181 (d) 173
23. 23.What is the smallest number you must add to 409 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
24. 24.What is the units digit of 8^57? (a) 2 (b) 8 (c) 6 (d) 4
25. 25.Which of these numbers is divisible by 8? (a) 1179 (b) 1170 (c) 1176 (d) 1186
26. 26.What is the units digit of 2^24? (a) 4 (b) 6 (c) 8 (d) 2
27. 27.What is the smallest number you must add to 172 to make it divisible by 3? (a) 2 (b) 1
28. 28.What is the smallest number you must add to 866 to make it divisible by 12? (a) 9 (b) 11 (c) 10 (d) 2
29. 29.What is the units digit of 3^53? (a) 1 (b) 7 (c) 3 (d) 9
30. 30.What is the units digit of 2^48? (a) 2 (b) 8 (c) 4 (d) 6
31. 31.What is the units digit of 2^38? (a) 4 (b) 2 (c) 8 (d) 6
32. 32.Which of these numbers is divisible by 4? (a) 1467 (b) 1464 (c) 1469 (d) 1463
33. 33.Which of these numbers is divisible by 6? (a) 1501 (b) 1497 (c) 1500 (d) 1507
34. 34.Which of these numbers is divisible by 9? (a) 1090 (b) 1080 (c) 1073 (d) 1088
35. 35.Which of these numbers is divisible by 4? (a) 2123 (b) 2124 (c) 2126 (d) 2131
36. 36.Which of these numbers is divisible by 3? (a) 975 (b) 974 (c) 979 (d) 977
37. 37.Which of these numbers is divisible by 3? (a) 2607 (b) 2605 (c) 2612 (d) 2609
38. 38.What is the remainder when 374 is divided by 7? (a) 5 (b) 2 (c) 4 (d) 3
39. 39.What is the smallest number you must add to 474 to make it divisible by 12? (a) 8 (b) 6 (c) 7 (d) 5
40. 40.What is the remainder when 253 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1

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