Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 868 · 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 4? (a) 2668 (b) 2675 (c) 2669 (d) 2667
2. 2.Which of these numbers is divisible by 9? (a) 7704 (b) 7707 (c) 7716 (d) 7701
3. 3.Which of these numbers is divisible by 8? (a) 2119 (b) 2102 (c) 2104 (d) 2108
4. 4.What is the units digit of 3^14? (a) 3 (b) 7 (c) 1 (d) 9
5. 5.What is the remainder when 249 is divided by 12? (a) 3 (b) 10 (c) 9 (d) 8
6. 6.Which of these numbers is divisible by 4? (a) 1392 (b) 1394 (c) 1397 (d) 1391
7. 7.Which of these numbers is divisible by 8? (a) 1535 (b) 1528 (c) 1542 (d) 1523
8. 8.Which of these numbers is divisible by 8? (a) 5193 (b) 5180 (c) 5184 (d) 5191
9. 9.Which of these numbers is divisible by 9? (a) 7830 (b) 7834 (c) 7829 (d) 7843
10. 10.What is the remainder when 267 is divided by 10? (a) 6 (b) 7 (c) 8 (d) 3
11. 11.Which of these numbers is divisible by 3? (a) 902 (b) 904 (c) 907 (d) 903
12. 12.What is the smallest number you must add to 375 to make it divisible by 11? (a) 8 (b) 1 (c) 9 (d) 10
13. 13.What is the smallest number you must add to 118 to make it divisible by 3? (a) 2 (b) 1
14. 14.What is the remainder when 627 is divided by 9? (a) 6 (b) 7 (c) 5 (d) 3
15. 15.What is the units digit of 8^11? (a) 6 (b) 8 (c) 4 (d) 2
16. 16.Which of these numbers is divisible by 6? (a) 5391 (b) 5385 (c) 5378 (d) 5382
17. 17.Which of these numbers is divisible by 8? (a) 4602 (b) 4611 (c) 4596 (d) 4600
18. 18.What is the smallest number you must add to 345 to make it divisible by 10? (a) 4 (b) 5 (c) 7 (d) 6
19. 19.What is the units digit of 8^21? (a) 8 (b) 4 (c) 2 (d) 6
20. 20.What is the remainder when 428 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
21. 21.Which of these numbers is divisible by 11? (a) 8712 (b) 8731 (c) 8705 (d) 8720
22. 22.What is the remainder when 752 is divided by 10? (a) 2 (b) 3 (c) 8 (d) 1
23. 23.What is the smallest number you must add to 195 to make it divisible by 10? (a) 7 (b) 5 (c) 4 (d) 6
24. 24.Which of these numbers is divisible by 9? (a) 5963 (b) 5982 (c) 5973 (d) 5967
25. 25.Which of these numbers is divisible by 4? (a) 799 (b) 807 (c) 800 (d) 801
26. 26.What is the units digit of 8^12? (a) 4 (b) 6 (c) 2 (d) 8
27. 27.What is the units digit of 4^21? (a) 3 (b) 6 (c) 4 (d) 5
28. 28.Which of these numbers is divisible by 8? (a) 1543 (b) 1551 (c) 1536 (d) 1530
29. 29.What is the remainder when 188 is divided by 3? (a) 0 (b) 1 (c) 2
30. 30.Which of these numbers is divisible by 9? (a) 6551 (b) 6557 (c) 6566 (d) 6552
31. 31.What is the smallest number you must add to 682 to make it divisible by 9? (a) 7 (b) 1 (c) 3 (d) 2
32. 32.Which of these numbers is divisible by 6? (a) 2222 (b) 2231 (c) 2215 (d) 2220
33. 33.What is the remainder when 73 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
34. 34.Which of these numbers is divisible by 9? (a) 2877 (b) 2862 (c) 2865 (d) 2856
35. 35.What is the remainder when 362 is divided by 12? (a) 2 (b) 1 (c) 3 (d) 10
36. 36.What is the smallest number you must add to 452 to make it divisible by 6? (a) 2 (b) 4 (c) 3 (d) 5
37. 37.What is the units digit of 9^32? (a) 1 (b) 0 (c) 9 (d) 2
38. 38.Which of these numbers is divisible by 4? (a) 436 (b) 439 (c) 442 (d) 433
39. 39.Which of these numbers is divisible by 9? (a) 6037 (b) 6024 (c) 6014 (d) 6021
40. 40.Which of these numbers is divisible by 6? (a) 996 (b) 998 (c) 1006 (d) 995

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