Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 374 · 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 8? (a) 2423 (b) 2424 (c) 2433 (d) 2429
2. 2.Which of these numbers is divisible by 8? (a) 1281 (b) 1288 (c) 1299 (d) 1291
3. 3.What is the units digit of 9^44? (a) 9 (b) 2 (c) 0 (d) 1
4. 4.What is the remainder when 394 is divided by 7? (a) 3 (b) 2 (c) 1 (d) 5
5. 5.What is the smallest number you must add to 31 to make it divisible by 3? (a) 1 (b) 2
6. 6.What is the smallest number you must add to 176 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
7. 7.Which of these numbers is divisible by 9? (a) 1769 (b) 1749 (c) 1755 (d) 1760
8. 8.Which of these numbers is divisible by 3? (a) 361 (b) 363 (c) 367 (d) 365
9. 9.What is the units digit of 3^15? (a) 1 (b) 9 (c) 7 (d) 3
10. 10.Which of these numbers is divisible by 8? (a) 5448 (b) 5442 (c) 5450 (d) 5461
11. 11.Which of these numbers is divisible by 8? (a) 744 (b) 759 (c) 746 (d) 740
12. 12.Which of these numbers is divisible by 3? (a) 240 (b) 239 (c) 245 (d) 242
13. 13.Which of these numbers is divisible by 3? (a) 462 (b) 467 (c) 460 (d) 464
14. 14.Which of these numbers is divisible by 11? (a) 3762 (b) 3753 (c) 3775 (d) 3772
15. 15.What is the remainder when 218 is divided by 7? (a) 6 (b) 1 (c) 2 (d) 0
16. 16.Which of these numbers is divisible by 9? (a) 5543 (b) 5534 (c) 5549 (d) 5535
17. 17.What is the smallest number you must add to 89 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
18. 18.What is the units digit of 7^18? (a) 7 (b) 1 (c) 9 (d) 3
19. 19.What is the remainder when 442 is divided by 11? (a) 1 (b) 3 (c) 9 (d) 2
20. 20.Which of these numbers is divisible by 6? (a) 3315 (b) 3305 (c) 3307 (d) 3306
21. 21.What is the smallest number you must add to 104 to make it divisible by 9? (a) 4 (b) 5 (c) 6 (d) 3
22. 22.What is the smallest number you must add to 373 to make it divisible by 10? (a) 6 (b) 3 (c) 8 (d) 7
23. 23.What is the remainder when 224 is divided by 3? (a) 1 (b) 2 (c) 0
24. 24.Which of these numbers is divisible by 4? (a) 1004 (b) 1005 (c) 1010 (d) 1001
25. 25.Which of these numbers is divisible by 11? (a) 3690 (b) 3684 (c) 3674 (d) 3668
26. 26.Which of these numbers is divisible by 3? (a) 620 (b) 625 (c) 621 (d) 622
27. 27.Which of these numbers is divisible by 8? (a) 5811 (b) 5817 (c) 5816 (d) 5829
28. 28.What is the smallest number you must add to 549 to make it divisible by 12? (a) 9 (b) 4 (c) 2 (d) 3
29. 29.What is the remainder when 272 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
30. 30.Which of these numbers is divisible by 3? (a) 356 (b) 358 (c) 362 (d) 357
31. 31.What is the smallest number you must add to 623 to make it divisible by 10? (a) 6 (b) 7 (c) 3 (d) 8
32. 32.What is the units digit of 2^46? (a) 6 (b) 4 (c) 2 (d) 8
33. 33.What is the remainder when 158 is divided by 7? (a) 6 (b) 5 (c) 3 (d) 4
34. 34.What is the remainder when 272 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 1
35. 35.Which of these numbers is divisible by 4? (a) 2596 (b) 2594 (c) 2597 (d) 2601
36. 36.Which of these numbers is divisible by 11? (a) 769 (b) 759 (c) 776 (d) 754
37. 37.What is the remainder when 102 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
38. 38.What is the remainder when 167 is divided by 8? (a) 1 (b) 5 (c) 7 (d) 6
39. 39.Which of these numbers is divisible by 8? (a) 6988 (b) 7005 (c) 6992 (d) 6996
40. 40.Which of these numbers is divisible by 8? (a) 4192 (b) 4203 (c) 4196 (d) 4186

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