Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 146 · 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 491 to make it divisible by 10? (a) 7 (b) 1 (c) 8 (d) 9
2. 2.Which of these numbers is divisible by 6? (a) 5292 (b) 5294 (c) 5288 (d) 5299
3. 3.What is the smallest number you must add to 152 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
4. 4.What is the remainder when 551 is divided by 7? (a) 6 (b) 2 (c) 5 (d) 4
5. 5.What is the smallest number you must add to 614 to make it divisible by 8? (a) 6 (b) 3 (c) 2 (d) 1
6. 6.Which of these numbers is divisible by 6? (a) 2682 (b) 2690 (c) 2687 (d) 2677
7. 7.What is the smallest number you must add to 178 to make it divisible by 3? (a) 2 (b) 1
8. 8.What is the smallest number you must add to 349 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 4
9. 9.What is the smallest number you must add to 80 to make it divisible by 3? (a) 2 (b) 1
10. 10.What is the remainder when 207 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
11. 11.Which of these numbers is divisible by 6? (a) 437 (b) 426 (c) 425 (d) 428
12. 12.Which of these numbers is divisible by 3? (a) 1565 (b) 1570 (c) 1566 (d) 1567
13. 13.What is the remainder when 956 is divided by 12? (a) 4 (b) 7 (c) 9 (d) 8
14. 14.What is the units digit of 3^31? (a) 3 (b) 7 (c) 9 (d) 1
15. 15.What is the remainder when 120 is divided by 7? (a) 6 (b) 2 (c) 1 (d) 0
16. 16.Which of these numbers is divisible by 4? (a) 626 (b) 622 (c) 630 (d) 624
17. 17.What is the smallest number you must add to 134 to make it divisible by 8? (a) 2 (b) 1 (c) 3 (d) 6
18. 18.What is the units digit of 7^22? (a) 3 (b) 9 (c) 1 (d) 7
19. 19.Which of these numbers is divisible by 3? (a) 1428 (b) 1433 (c) 1430 (d) 1427
20. 20.What is the units digit of 9^22? (a) 9 (b) 1 (c) 2 (d) 0
21. 21.What is the smallest number you must add to 94 to make it divisible by 3? (a) 2 (b) 1
22. 22.What is the units digit of 2^59? (a) 6 (b) 8 (c) 4 (d) 2
23. 23.Which of these numbers is divisible by 9? (a) 4575 (b) 4563 (c) 4560 (d) 4565
24. 24.What is the units digit of 2^34? (a) 8 (b) 2 (c) 6 (d) 4
25. 25.What is the smallest number you must add to 642 to make it divisible by 8? (a) 6 (b) 5 (c) 2 (d) 7
26. 26.Which of these numbers is divisible by 3? (a) 543 (b) 547 (c) 542 (d) 544
27. 27.What is the units digit of 2^32? (a) 4 (b) 8 (c) 2 (d) 6
28. 28.Which of these numbers is divisible by 8? (a) 4248 (b) 4254 (c) 4246 (d) 4259
29. 29.Which of these numbers is divisible by 3? (a) 1696 (b) 1693 (c) 1692 (d) 1691
30. 30.Which of these numbers is divisible by 3? (a) 1726 (b) 1724 (c) 1722 (d) 1721
31. 31.What is the units digit of 4^45? (a) 3 (b) 4 (c) 6 (d) 5
32. 32.What is the remainder when 503 is divided by 7? (a) 6 (b) 4 (c) 1 (d) 5
33. 33.What is the units digit of 3^44? (a) 1 (b) 7 (c) 3 (d) 9
34. 34.What is the units digit of 7^42? (a) 1 (b) 9 (c) 3 (d) 7
35. 35.Which of these numbers is divisible by 11? (a) 1322 (b) 1309 (c) 1299 (d) 1312
36. 36.What is the smallest number you must add to 159 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
37. 37.What is the units digit of 9^60? (a) 9 (b) 0 (c) 1 (d) 2
38. 38.What is the smallest number you must add to 249 to make it divisible by 7? (a) 5 (b) 4 (c) 2 (d) 3
39. 39.What is the smallest number you must add to 741 to make it divisible by 10? (a) 1 (b) 9 (c) 7 (d) 8
40. 40.Which of these numbers is divisible by 11? (a) 1710 (b) 1701 (c) 1720 (d) 1705

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