Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 643 · 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 3^35? (a) 3 (b) 9 (c) 1 (d) 7
2. 2.What is the smallest number you must add to 375 to make it divisible by 7? (a) 2 (b) 3 (c) 5 (d) 4
3. 3.Which of these numbers is divisible by 8? (a) 4304 (b) 4307 (c) 4299 (d) 4315
4. 4.What is the smallest number you must add to 583 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 7
5. 5.What is the units digit of 2^39? (a) 8 (b) 4 (c) 6 (d) 2
6. 6.What is the smallest number you must add to 139 to make it divisible by 7? (a) 1 (b) 6 (c) 3 (d) 2
7. 7.Which of these numbers is divisible by 8? (a) 6455 (b) 6445 (c) 6458 (d) 6448
8. 8.What is the remainder when 215 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
9. 9.Which of these numbers is divisible by 9? (a) 5727 (b) 5715 (c) 5720 (d) 5710
10. 10.Which of these numbers is divisible by 9? (a) 1746 (b) 1751 (c) 1757 (d) 1741
11. 11.What is the units digit of 7^58? (a) 3 (b) 1 (c) 9 (d) 7
12. 12.What is the smallest number you must add to 326 to make it divisible by 7? (a) 3 (b) 5 (c) 4 (d) 2
13. 13.Which of these numbers is divisible by 3? (a) 2401 (b) 2399 (c) 2404 (d) 2400
14. 14.What is the units digit of 9^19? (a) 7 (b) 9 (c) 1 (d) 8
15. 15.What is the remainder when 281 is divided by 8? (a) 2 (b) 7 (c) 0 (d) 1
16. 16.What is the remainder when 40 is divided by 3? (a) 0 (b) 1 (c) 2
17. 17.What is the smallest number you must add to 137 to make it divisible by 3? (a) 2 (b) 1
18. 18.Which of these numbers is divisible by 11? (a) 4312 (b) 4308 (c) 4319 (d) 4332
19. 19.What is the smallest number you must add to 355 to make it divisible by 7? (a) 2 (b) 1 (c) 3 (d) 5
20. 20.What is the remainder when 932 is divided by 12? (a) 7 (b) 9 (c) 8 (d) 4
21. 21.Which of these numbers is divisible by 9? (a) 3930 (b) 3919 (c) 3924 (d) 3936
22. 22.What is the units digit of 3^31? (a) 7 (b) 9 (c) 3 (d) 1
23. 23.Which of these numbers is divisible by 11? (a) 9240 (b) 9236 (c) 9261 (d) 9243
24. 24.What is the remainder when 233 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
25. 25.What is the smallest number you must add to 208 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 2
26. 26.What is the units digit of 4^57? (a) 5 (b) 6 (c) 3 (d) 4
27. 27.Which of these numbers is divisible by 8? (a) 1456 (b) 1471 (c) 1453 (d) 1458
28. 28.Which of these numbers is divisible by 9? (a) 1734 (b) 1727 (c) 1728 (d) 1744
29. 29.Which of these numbers is divisible by 11? (a) 5524 (b) 5520 (c) 5522 (d) 5537
30. 30.Which of these numbers is divisible by 4? (a) 1741 (b) 1734 (c) 1736 (d) 1737
31. 31.What is the remainder when 519 is divided by 11? (a) 1 (b) 9 (c) 3 (d) 2
32. 32.What is the remainder when 235 is divided by 10? (a) 6 (b) 5 (c) 7 (d) 4
33. 33.What is the smallest number you must add to 85 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 5
34. 34.What is the remainder when 353 is divided by 12? (a) 7 (b) 5 (c) 6 (d) 4
35. 35.What is the units digit of 2^37? (a) 8 (b) 4 (c) 6 (d) 2
36. 36.What is the remainder when 596 is divided by 9? (a) 2 (b) 7 (c) 1 (d) 3
37. 37.What is the smallest number you must add to 401 to make it divisible by 8? (a) 1 (b) 7 (c) 5 (d) 6
38. 38.What is the smallest number you must add to 256 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 1
39. 39.What is the smallest number you must add to 62 to make it divisible by 3? (a) 1 (b) 2
40. 40.Which of these numbers is divisible by 9? (a) 3462 (b) 3446 (c) 3447 (d) 3452

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