Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 128 · 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 smallest number you must add to 54 to make it divisible by 4? (a) 2 (b) 1 (c) 3
2. 2.What is the remainder when 370 is divided by 9? (a) 1 (b) 8 (c) 0 (d) 2
3. 3.What is the units digit of 3^42? (a) 7 (b) 1 (c) 9 (d) 3
4. 4.Which of these numbers is divisible by 6? (a) 2917 (b) 2915 (c) 2923 (d) 2916
5. 5.What is the smallest number you must add to 256 to make it divisible by 9? (a) 5 (b) 7 (c) 6 (d) 4
6. 6.Which of these numbers is divisible by 6? (a) 4661 (b) 4652 (c) 4666 (d) 4656
7. 7.What is the units digit of 3^43? (a) 9 (b) 3 (c) 1 (d) 7
8. 8.Which of these numbers is divisible by 6? (a) 4841 (b) 4831 (c) 4830 (d) 4828
9. 9.Which of these numbers is divisible by 3? (a) 633 (b) 637 (c) 632 (d) 634
10. 10.Which of these numbers is divisible by 4? (a) 3586 (b) 3591 (c) 3584 (d) 3583
11. 11.What is the units digit of 7^21? (a) 9 (b) 3 (c) 7 (d) 1
12. 12.What is the remainder when 670 is divided by 9? (a) 4 (b) 5 (c) 3 (d) 6
13. 13.What is the units digit of 4^12? (a) 4 (b) 7 (c) 5 (d) 6
14. 14.What is the smallest number you must add to 274 to make it divisible by 4? (a) 3 (b) 2 (c) 1
15. 15.Which of these numbers is divisible by 9? (a) 1793 (b) 1800 (c) 1806 (d) 1813
16. 16.What is the units digit of 2^23? (a) 8 (b) 4 (c) 6 (d) 2
17. 17.What is the units digit of 3^30? (a) 7 (b) 1 (c) 9 (d) 3
18. 18.What is the units digit of 9^56? (a) 0 (b) 1 (c) 9 (d) 2
19. 19.Which of these numbers is divisible by 8? (a) 2611 (b) 2616 (c) 2629 (d) 2621
20. 20.What is the units digit of 8^24? (a) 4 (b) 6 (c) 8 (d) 2
21. 21.Which of these numbers is divisible by 11? (a) 2778 (b) 2787 (c) 2795 (d) 2783
22. 22.What is the units digit of 4^24? (a) 4 (b) 6 (c) 5 (d) 7
23. 23.Which of these numbers is divisible by 3? (a) 1042 (b) 1049 (c) 1044 (d) 1045
24. 24.Which of these numbers is divisible by 8? (a) 5858 (b) 5850 (c) 5871 (d) 5856
25. 25.Which of these numbers is divisible by 4? (a) 3404 (b) 3409 (c) 3406 (d) 3402
26. 26.What is the remainder when 68 is divided by 6? (a) 2 (b) 4 (c) 1 (d) 3
27. 27.What is the units digit of 9^51? (a) 1 (b) 8 (c) 9 (d) 7
28. 28.What is the units digit of 7^50? (a) 7 (b) 3 (c) 9 (d) 1
29. 29.What is the remainder when 216 is divided by 10? (a) 5 (b) 4 (c) 6 (d) 7
30. 30.What is the units digit of 9^13? (a) 9 (b) 1 (c) 7 (d) 8
31. 31.What is the units digit of 8^19? (a) 4 (b) 6 (c) 8 (d) 2
32. 32.What is the remainder when 243 is divided by 7? (a) 6 (b) 2 (c) 4 (d) 5
33. 33.What is the units digit of 2^41? (a) 2 (b) 6 (c) 4 (d) 8
34. 34.What is the units digit of 2^10? (a) 6 (b) 2 (c) 8 (d) 4
35. 35.What is the remainder when 173 is divided by 8? (a) 6 (b) 5 (c) 4 (d) 3
36. 36.Which of these numbers is divisible by 6? (a) 282 (b) 284 (c) 280 (d) 292
37. 37.Which of these numbers is divisible by 4? (a) 2173 (b) 2179 (c) 2172 (d) 2171
38. 38.What is the remainder when 263 is divided by 7? (a) 5 (b) 3 (c) 4 (d) 6
39. 39.What is the remainder when 93 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
40. 40.What is the units digit of 2^17? (a) 4 (b) 2 (c) 8 (d) 6

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