Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 567 · 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 6? (a) 773 (b) 766 (c) 768 (d) 775
2. 2.What is the smallest number you must add to 566 to make it divisible by 11? (a) 7 (b) 6 (c) 8 (d) 5
3. 3.Which of these numbers is divisible by 9? (a) 1697 (b) 1705 (c) 1690 (d) 1692
4. 4.What is the remainder when 346 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
5. 5.What is the units digit of 2^20? (a) 8 (b) 6 (c) 2 (d) 4
6. 6.What is the smallest number you must add to 94 to make it divisible by 4? (a) 2 (b) 3 (c) 1
7. 7.What is the smallest number you must add to 233 to make it divisible by 4? (a) 3 (b) 2 (c) 1
8. 8.What is the smallest number you must add to 227 to make it divisible by 9? (a) 7 (b) 8 (c) 6 (d) 2
9. 9.What is the remainder when 138 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
10. 10.What is the remainder when 95 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
11. 11.Which of these numbers is divisible by 3? (a) 1082 (b) 1080 (c) 1079 (d) 1085
12. 12.Which of these numbers is divisible by 8? (a) 3725 (b) 3717 (c) 3734 (d) 3720
13. 13.Which of these numbers is divisible by 6? (a) 534 (b) 529 (c) 536 (d) 541
14. 14.What is the units digit of 3^52? (a) 9 (b) 7 (c) 3 (d) 1
15. 15.Which of these numbers is divisible by 3? (a) 1733 (b) 1730 (c) 1728 (d) 1727
16. 16.What is the units digit of 2^30? (a) 4 (b) 8 (c) 6 (d) 2
17. 17.Which of these numbers is divisible by 4? (a) 2853 (b) 2859 (c) 2851 (d) 2852
18. 18.What is the smallest number you must add to 54 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
19. 19.Which of these numbers is divisible by 4? (a) 1739 (b) 1743 (c) 1745 (d) 1740
20. 20.What is the units digit of 9^57? (a) 1 (b) 8 (c) 7 (d) 9
21. 21.What is the smallest number you must add to 457 to make it divisible by 10? (a) 3 (b) 4 (c) 7 (d) 2
22. 22.What is the smallest number you must add to 142 to make it divisible by 10? (a) 7 (b) 8 (c) 2 (d) 9
23. 23.What is the remainder when 62 is divided by 3? (a) 0 (b) 1 (c) 2
24. 24.What is the units digit of 7^53? (a) 3 (b) 1 (c) 9 (d) 7
25. 25.What is the smallest number you must add to 270 to make it divisible by 4? (a) 2 (b) 1 (c) 3
26. 26.What is the smallest number you must add to 312 to make it divisible by 9? (a) 6 (b) 2 (c) 3 (d) 4
27. 27.What is the smallest number you must add to 606 to make it divisible by 9? (a) 3 (b) 6 (c) 5 (d) 7
28. 28.Which of these numbers is divisible by 3? (a) 578 (b) 574 (c) 572 (d) 573
29. 29.What is the units digit of 8^43? (a) 8 (b) 4 (c) 2 (d) 6
30. 30.Which of these numbers is divisible by 8? (a) 1173 (b) 1155 (c) 1160 (d) 1161
31. 31.What is the units digit of 4^21? (a) 4 (b) 5 (c) 3 (d) 6
32. 32.Which of these numbers is divisible by 8? (a) 5532 (b) 5538 (c) 5523 (d) 5528
33. 33.What is the smallest number you must add to 251 to make it divisible by 10? (a) 8 (b) 7 (c) 1 (d) 9
34. 34.What is the units digit of 3^29? (a) 1 (b) 3 (c) 9 (d) 7
35. 35.What is the units digit of 4^26? (a) 5 (b) 7 (c) 4 (d) 6
36. 36.Which of these numbers is divisible by 6? (a) 4011 (b) 4002 (c) 4004 (d) 3999
37. 37.What is the remainder when 112 is divided by 3? (a) 0 (b) 2 (c) 1
38. 38.What is the units digit of 3^31? (a) 3 (b) 9 (c) 7 (d) 1
39. 39.Which of these numbers is divisible by 9? (a) 3273 (b) 3262 (c) 3258 (d) 3252
40. 40.What is the units digit of 2^32? (a) 4 (b) 2 (c) 8 (d) 6

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