Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 498 · 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 remainder when 599 is divided by 9? (a) 7 (b) 5 (c) 6 (d) 4
2. 2.Which of these numbers is divisible by 9? (a) 5435 (b) 5418 (c) 5411 (d) 5424
3. 3.What is the units digit of 8^24? (a) 2 (b) 8 (c) 6 (d) 4
4. 4.What is the smallest number you must add to 177 to make it divisible by 7? (a) 6 (b) 4 (c) 2 (d) 5
5. 5.What is the units digit of 9^31? (a) 1 (b) 7 (c) 8 (d) 9
6. 6.What is the remainder when 193 is divided by 9? (a) 5 (b) 4 (c) 3 (d) 6
7. 7.Which of these numbers is divisible by 8? (a) 5733 (b) 5721 (c) 5720 (d) 5715
8. 8.What is the smallest number you must add to 338 to make it divisible by 9? (a) 3 (b) 5 (c) 4 (d) 6
9. 9.What is the remainder when 622 is divided by 9? (a) 2 (b) 8 (c) 0 (d) 1
10. 10.Which of these numbers is divisible by 11? (a) 3744 (b) 3737 (c) 3729 (d) 3727
11. 11.Which of these numbers is divisible by 8? (a) 6312 (b) 6325 (c) 6315 (d) 6306
12. 12.What is the remainder when 110 is divided by 3? (a) 0 (b) 1 (c) 2
13. 13.What is the remainder when 654 is divided by 9? (a) 6 (b) 5 (c) 7 (d) 3
14. 14.What is the units digit of 9^27? (a) 1 (b) 8 (c) 9 (d) 7
15. 15.What is the smallest number you must add to 210 to make it divisible by 4? (a) 1 (b) 3 (c) 2
16. 16.What is the remainder when 399 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
17. 17.What is the smallest number you must add to 107 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
18. 18.Which of these numbers is divisible by 9? (a) 5616 (b) 5610 (c) 5624 (d) 5629
19. 19.What is the units digit of 4^49? (a) 6 (b) 4 (c) 5 (d) 3
20. 20.What is the smallest number you must add to 97 to make it divisible by 4? (a) 1 (b) 3 (c) 2
21. 21.Which of these numbers is divisible by 6? (a) 1581 (b) 1584 (c) 1587 (d) 1594
22. 22.What is the units digit of 7^31? (a) 1 (b) 7 (c) 3 (d) 9
23. 23.What is the remainder when 71 is divided by 6? (a) 3 (b) 5 (c) 1 (d) 4
24. 24.What is the smallest number you must add to 149 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 2
25. 25.What is the remainder when 164 is divided by 3? (a) 2 (b) 0 (c) 1
26. 26.What is the units digit of 2^30? (a) 4 (b) 6 (c) 8 (d) 2
27. 27.Which of these numbers is divisible by 6? (a) 4750 (b) 4743 (c) 4753 (d) 4746
28. 28.Which of these numbers is divisible by 4? (a) 3109 (b) 3117 (c) 3114 (d) 3112
29. 29.What is the smallest number you must add to 757 to make it divisible by 11? (a) 9 (b) 2 (c) 1 (d) 3
30. 30.What is the smallest number you must add to 325 to make it divisible by 9? (a) 1 (b) 6 (c) 7 (d) 8
31. 31.What is the units digit of 7^41? (a) 3 (b) 1 (c) 9 (d) 7
32. 32.Which of these numbers is divisible by 8? (a) 6737 (b) 6750 (c) 6736 (d) 6733
33. 33.What is the remainder when 122 is divided by 10? (a) 3 (b) 8 (c) 1 (d) 2
34. 34.Which of these numbers is divisible by 8? (a) 2216 (b) 2228 (c) 2210 (d) 2222
35. 35.What is the remainder when 865 is divided by 11? (a) 7 (b) 6 (c) 4 (d) 8
36. 36.What is the remainder when 808 is divided by 10? (a) 8 (b) 2 (c) 7 (d) 9
37. 37.Which of these numbers is divisible by 8? (a) 1463 (b) 1474 (c) 1466 (d) 1464
38. 38.Which of these numbers is divisible by 3? (a) 1571 (b) 1572 (c) 1577 (d) 1573
39. 39.What is the units digit of 4^54? (a) 6 (b) 7 (c) 5 (d) 4
40. 40.What is the smallest number you must add to 332 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3

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