Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 967 · 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 smallest number you must add to 645 to make it divisible by 8? (a) 5 (b) 2 (c) 4 (d) 3
2. 2.What is the remainder when 873 is divided by 11? (a) 7 (b) 5 (c) 4 (d) 3
3. 3.Which of these numbers is divisible by 9? (a) 3313 (b) 3324 (c) 3312 (d) 3310
4. 4.What is the remainder when 482 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
5. 5.What is the units digit of 3^20? (a) 7 (b) 9 (c) 3 (d) 1
6. 6.What is the smallest number you must add to 673 to make it divisible by 10? (a) 8 (b) 6 (c) 7 (d) 3
7. 7.Which of these numbers is divisible by 3? (a) 2446 (b) 2441 (c) 2442 (d) 2444
8. 8.What is the smallest number you must add to 329 to make it divisible by 6? (a) 2 (b) 1 (c) 5 (d) 3
9. 9.What is the units digit of 7^10? (a) 3 (b) 9 (c) 1 (d) 7
10. 10.Which of these numbers is divisible by 4? (a) 2588 (b) 2587 (c) 2591 (d) 2594
11. 11.Which of these numbers is divisible by 4? (a) 1975 (b) 1968 (c) 1967 (d) 1970
12. 12.Which of these numbers is divisible by 9? (a) 6675 (b) 6694 (c) 6683 (d) 6678
13. 13.Which of these numbers is divisible by 6? (a) 597 (b) 600 (c) 607 (d) 603
14. 14.What is the smallest number you must add to 77 to make it divisible by 4? (a) 3 (b) 1 (c) 2
15. 15.Which of these numbers is divisible by 4? (a) 1771 (b) 1779 (c) 1775 (d) 1772
16. 16.What is the smallest number you must add to 612 to make it divisible by 10? (a) 9 (b) 8 (c) 2 (d) 7
17. 17.What is the smallest number you must add to 307 to make it divisible by 4? (a) 3 (b) 1 (c) 2
18. 18.What is the units digit of 7^29? (a) 3 (b) 1 (c) 9 (d) 7
19. 19.Which of these numbers is divisible by 11? (a) 7010 (b) 7018 (c) 7039 (d) 7026
20. 20.What is the smallest number you must add to 576 to make it divisible by 11? (a) 4 (b) 8 (c) 7 (d) 6
21. 21.Which of these numbers is divisible by 11? (a) 6508 (b) 6483 (c) 6493 (d) 6490
22. 22.What is the units digit of 3^11? (a) 9 (b) 3 (c) 1 (d) 7
23. 23.Which of these numbers is divisible by 8? (a) 5143 (b) 5151 (c) 5144 (d) 5159
24. 24.Which of these numbers is divisible by 8? (a) 3842 (b) 3848 (c) 3850 (d) 3863
25. 25.Which of these numbers is divisible by 6? (a) 1503 (b) 1495 (c) 1494 (d) 1493
26. 26.Which of these numbers is divisible by 3? (a) 396 (b) 401 (c) 395 (d) 397
27. 27.Which of these numbers is divisible by 11? (a) 8829 (b) 8835 (c) 8833 (d) 8854
28. 28.What is the remainder when 506 is divided by 12? (a) 1 (b) 3 (c) 2 (d) 10
29. 29.Which of these numbers is divisible by 11? (a) 9665 (b) 9675 (c) 9689 (d) 9669
30. 30.What is the smallest number you must add to 640 to make it divisible by 9? (a) 6 (b) 8 (c) 1 (d) 7
31. 31.Which of these numbers is divisible by 11? (a) 2120 (b) 2128 (c) 2138 (d) 2123
32. 32.What is the remainder when 521 is divided by 7? (a) 4 (b) 3 (c) 5 (d) 2
33. 33.What is the units digit of 3^23? (a) 9 (b) 7 (c) 1 (d) 3
34. 34.What is the remainder when 167 is divided by 3? (a) 0 (b) 2 (c) 1
35. 35.What is the smallest number you must add to 156 to make it divisible by 11? (a) 2 (b) 10 (c) 9 (d) 8
36. 36.What is the remainder when 256 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
37. 37.What is the smallest number you must add to 493 to make it divisible by 8? (a) 3 (b) 2 (c) 5 (d) 4
38. 38.Which of these numbers is divisible by 4? (a) 1542 (b) 1536 (c) 1538 (d) 1535
39. 39.What is the units digit of 2^52? (a) 6 (b) 8 (c) 4 (d) 2
40. 40.What is the units digit of 7^38? (a) 9 (b) 3 (c) 1 (d) 7

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