Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 845 · 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 173 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
2. 2.Which of these numbers is divisible by 3? (a) 2528 (b) 2531 (c) 2526 (d) 2525
3. 3.What is the remainder when 156 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
4. 4.What is the remainder when 73 is divided by 6? (a) 2 (b) 0 (c) 5 (d) 1
5. 5.What is the smallest number you must add to 586 to make it divisible by 10? (a) 3 (b) 5 (c) 6 (d) 4
6. 6.Which of these numbers is divisible by 3? (a) 2680 (b) 2679 (c) 2683 (d) 2678
7. 7.What is the remainder when 120 is divided by 11? (a) 1 (b) 8 (c) 10 (d) 9
8. 8.What is the remainder when 114 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
9. 9.Which of these numbers is divisible by 4? (a) 3250 (b) 3245 (c) 3254 (d) 3248
10. 10.What is the units digit of 7^44? (a) 9 (b) 3 (c) 7 (d) 1
11. 11.Which of these numbers is divisible by 9? (a) 2545 (b) 2558 (c) 2547 (d) 2551
12. 12.Which of these numbers is divisible by 3? (a) 630 (b) 632 (c) 634 (d) 628
13. 13.What is the remainder when 257 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
14. 14.What is the remainder when 307 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
15. 15.What is the units digit of 4^19? (a) 4 (b) 6 (c) 5 (d) 3
16. 16.Which of these numbers is divisible by 6? (a) 1613 (b) 1614 (c) 1621 (d) 1618
17. 17.Which of these numbers is divisible by 4? (a) 928 (b) 933 (c) 926 (d) 929
18. 18.What is the units digit of 3^34? (a) 1 (b) 3 (c) 9 (d) 7
19. 19.What is the remainder when 135 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 7
20. 20.What is the remainder when 310 is divided by 6? (a) 2 (b) 5 (c) 3 (d) 4
21. 21.Which of these numbers is divisible by 11? (a) 6597 (b) 6589 (c) 6581 (d) 6604
22. 22.What is the units digit of 7^10? (a) 7 (b) 1 (c) 9 (d) 3
23. 23.Which of these numbers is divisible by 11? (a) 9057 (b) 9043 (c) 9035 (d) 9042
24. 24.Which of these numbers is divisible by 11? (a) 1319 (b) 1333 (c) 1320 (d) 1326
25. 25.What is the smallest number you must add to 167 to make it divisible by 3? (a) 2 (b) 1
26. 26.What is the units digit of 4^45? (a) 6 (b) 5 (c) 3 (d) 4
27. 27.Which of these numbers is divisible by 8? (a) 4592 (b) 4595 (c) 4605 (d) 4586
28. 28.What is the remainder when 133 is divided by 3? (a) 0 (b) 2 (c) 1
29. 29.What is the units digit of 2^21? (a) 6 (b) 2 (c) 8 (d) 4
30. 30.What is the smallest number you must add to 256 to make it divisible by 12? (a) 7 (b) 8 (c) 4 (d) 9
31. 31.What is the remainder when 820 is divided by 12? (a) 3 (b) 4 (c) 5 (d) 8
32. 32.Which of these numbers is divisible by 8? (a) 1684 (b) 1688 (c) 1694 (d) 1698
33. 33.What is the remainder when 85 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
34. 34.What is the smallest number you must add to 425 to make it divisible by 12? (a) 7 (b) 5 (c) 6 (d) 8
35. 35.Which of these numbers is divisible by 9? (a) 5073 (b) 5058 (c) 5053 (d) 5060
36. 36.What is the remainder when 61 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
37. 37.What is the smallest number you must add to 362 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
38. 38.Which of these numbers is divisible by 4? (a) 2687 (b) 2683 (c) 2689 (d) 2684
39. 39.What is the units digit of 9^50? (a) 2 (b) 0 (c) 9 (d) 1
40. 40.What is the units digit of 4^37? (a) 4 (b) 6 (c) 5 (d) 3

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