Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 976 · 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 148 is divided by 3? (a) 2 (b) 1 (c) 0
2. 2.Which of these numbers is divisible by 4? (a) 1453 (b) 1445 (c) 1449 (d) 1448
3. 3.What is the smallest number you must add to 443 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 2
4. 4.What is the units digit of 3^42? (a) 3 (b) 1 (c) 9 (d) 7
5. 5.What is the smallest number you must add to 156 to make it divisible by 10? (a) 3 (b) 5 (c) 6 (d) 4
6. 6.Which of these numbers is divisible by 6? (a) 1738 (b) 1743 (c) 1733 (d) 1734
7. 7.What is the units digit of 8^23? (a) 4 (b) 2 (c) 8 (d) 6
8. 8.What is the remainder when 241 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
9. 9.Which of these numbers is divisible by 6? (a) 4508 (b) 4500 (c) 4499 (d) 4504
10. 10.What is the remainder when 31 is divided by 3? (a) 1 (b) 0 (c) 2
11. 11.What is the units digit of 7^43? (a) 3 (b) 7 (c) 9 (d) 1
12. 12.Which of these numbers is divisible by 4? (a) 1957 (b) 1953 (c) 1962 (d) 1956
13. 13.What is the smallest number you must add to 117 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
14. 14.Which of these numbers is divisible by 6? (a) 902 (b) 898 (c) 894 (d) 893
15. 15.Which of these numbers is divisible by 3? (a) 1824 (b) 1825 (c) 1823 (d) 1829
16. 16.What is the units digit of 4^32? (a) 5 (b) 6 (c) 4 (d) 7
17. 17.What is the smallest number you must add to 274 to make it divisible by 4? (a) 1 (b) 3 (c) 2
18. 18.What is the units digit of 4^52? (a) 4 (b) 6 (c) 5 (d) 7
19. 19.What is the smallest number you must add to 108 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
20. 20.What is the units digit of 2^14? (a) 8 (b) 4 (c) 6 (d) 2
21. 21.Which of these numbers is divisible by 8? (a) 5143 (b) 5132 (c) 5128 (d) 5121
22. 22.What is the smallest number you must add to 205 to make it divisible by 4? (a) 3 (b) 2 (c) 1
23. 23.What is the remainder when 318 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
24. 24.What is the remainder when 235 is divided by 9? (a) 8 (b) 2 (c) 0 (d) 1
25. 25.Which of these numbers is divisible by 3? (a) 857 (b) 859 (c) 854 (d) 855
26. 26.Which of these numbers is divisible by 8? (a) 6132 (b) 6120 (c) 6119 (d) 6127
27. 27.What is the smallest number you must add to 359 to make it divisible by 7? (a) 2 (b) 6 (c) 4 (d) 5
28. 28.What is the remainder when 702 is divided by 12? (a) 7 (b) 8 (c) 5 (d) 6
29. 29.Which of these numbers is divisible by 4? (a) 1121 (b) 1126 (c) 1129 (d) 1124
30. 30.What is the smallest number you must add to 231 to make it divisible by 4? (a) 2 (b) 3 (c) 1
31. 31.What is the smallest number you must add to 603 to make it divisible by 11? (a) 3 (b) 1 (c) 2 (d) 9
32. 32.What is the units digit of 2^60? (a) 2 (b) 4 (c) 8 (d) 6
33. 33.What is the remainder when 268 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
34. 34.What is the units digit of 8^15? (a) 6 (b) 4 (c) 8 (d) 2
35. 35.What is the smallest number you must add to 631 to make it divisible by 8? (a) 7 (b) 3 (c) 1 (d) 2
36. 36.What is the units digit of 8^60? (a) 2 (b) 8 (c) 6 (d) 4
37. 37.Which of these numbers is divisible by 11? (a) 6771 (b) 6792 (c) 6776 (d) 6778
38. 38.Which of these numbers is divisible by 3? (a) 1689 (b) 1694 (c) 1690 (d) 1687
39. 39.What is the remainder when 65 is divided by 3? (a) 0 (b) 1 (c) 2
40. 40.What is the remainder when 125 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3

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