Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 442 · 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) 1047 (b) 1037 (c) 1042 (d) 1040
2. 2.What is the remainder when 478 is divided by 8? (a) 6 (b) 2 (c) 5 (d) 7
3. 3.Which of these numbers is divisible by 4? (a) 776 (b) 775 (c) 782 (d) 779
4. 4.Which of these numbers is divisible by 8? (a) 4428 (b) 4420 (c) 4424 (d) 4436
5. 5.What is the remainder when 223 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
6. 6.What is the units digit of 9^39? (a) 8 (b) 1 (c) 9 (d) 7
7. 7.Which of these numbers is divisible by 6? (a) 3949 (b) 3957 (c) 3948 (d) 3947
8. 8.What is the remainder when 276 is divided by 9? (a) 3 (b) 6 (c) 7 (d) 5
9. 9.What is the remainder when 448 is divided by 10? (a) 7 (b) 9 (c) 8 (d) 2
10. 10.What is the smallest number you must add to 824 to make it divisible by 11? (a) 3 (b) 10 (c) 2 (d) 1
11. 11.What is the remainder when 143 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
12. 12.Which of these numbers is divisible by 9? (a) 6983 (b) 6990 (c) 6974 (d) 6975
13. 13.Which of these numbers is divisible by 8? (a) 1336 (b) 1331 (c) 1337 (d) 1347
14. 14.Which of these numbers is divisible by 6? (a) 1201 (b) 1195 (c) 1208 (d) 1200
15. 15.What is the units digit of 3^10? (a) 3 (b) 1 (c) 7 (d) 9
16. 16.What is the units digit of 2^23? (a) 2 (b) 8 (c) 4 (d) 6
17. 17.Which of these numbers is divisible by 9? (a) 4218 (b) 4229 (c) 4204 (d) 4212
18. 18.What is the smallest number you must add to 376 to make it divisible by 11? (a) 2 (b) 10 (c) 9 (d) 8
19. 19.What is the smallest number you must add to 321 to make it divisible by 11? (a) 9 (b) 10 (c) 2 (d) 8
20. 20.Which of these numbers is divisible by 11? (a) 7006 (b) 7022 (c) 7007 (d) 7009
21. 21.Which of these numbers is divisible by 9? (a) 757 (b) 765 (c) 782 (d) 772
22. 22.What is the smallest number you must add to 311 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 7
23. 23.Which of these numbers is divisible by 3? (a) 2371 (b) 2368 (c) 2370 (d) 2375
24. 24.What is the smallest number you must add to 175 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 4
25. 25.What is the remainder when 279 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
26. 26.What is the remainder when 49 is divided by 3? (a) 1 (b) 2 (c) 0
27. 27.Which of these numbers is divisible by 6? (a) 1672 (b) 1677 (c) 1668 (d) 1667
28. 28.Which of these numbers is divisible by 3? (a) 1978 (b) 1974 (c) 1972 (d) 1976
29. 29.What is the remainder when 122 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
30. 30.What is the units digit of 9^20? (a) 9 (b) 2 (c) 1 (d) 0
31. 31.What is the smallest number you must add to 71 to make it divisible by 3? (a) 1 (b) 2
32. 32.What is the units digit of 2^25? (a) 6 (b) 4 (c) 8 (d) 2
33. 33.Which of these numbers is divisible by 3? (a) 2168 (b) 2164 (c) 2170 (d) 2166
34. 34.Which of these numbers is divisible by 9? (a) 1637 (b) 1645 (c) 1654 (d) 1638
35. 35.What is the smallest number you must add to 533 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
36. 36.What is the units digit of 9^28? (a) 0 (b) 9 (c) 2 (d) 1
37. 37.Which of these numbers is divisible by 6? (a) 3722 (b) 3709 (c) 3714 (d) 3715
38. 38.What is the smallest number you must add to 299 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 2
39. 39.Which of these numbers is divisible by 4? (a) 314 (b) 308 (c) 310 (d) 307
40. 40.Which of these numbers is divisible by 9? (a) 7661 (b) 7648 (c) 7657 (d) 7650

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