Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 709 · 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 174 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
2. 2.What is the units digit of 2^43? (a) 6 (b) 8 (c) 4 (d) 2
3. 3.What is the units digit of 8^25? (a) 6 (b) 8 (c) 2 (d) 4
4. 4.What is the smallest number you must add to 655 to make it divisible by 12? (a) 6 (b) 4 (c) 7 (d) 5
5. 5.Which of these numbers is divisible by 9? (a) 8045 (b) 8061 (c) 8047 (d) 8046
6. 6.Which of these numbers is divisible by 4? (a) 2981 (b) 2973 (c) 2977 (d) 2976
7. 7.What is the units digit of 7^29? (a) 7 (b) 9 (c) 1 (d) 3
8. 8.Which of these numbers is divisible by 6? (a) 855 (b) 858 (c) 866 (d) 859
9. 9.What is the remainder when 95 is divided by 3? (a) 1 (b) 0 (c) 2
10. 10.What is the remainder when 124 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
11. 11.What is the units digit of 4^18? (a) 4 (b) 6 (c) 7 (d) 5
12. 12.What is the remainder when 246 is divided by 12? (a) 6 (b) 7 (c) 5 (d) 8
13. 13.What is the units digit of 8^18? (a) 4 (b) 8 (c) 2 (d) 6
14. 14.Which of these numbers is divisible by 4? (a) 3395 (b) 3392 (c) 3389 (d) 3399
15. 15.What is the units digit of 2^41? (a) 4 (b) 6 (c) 8 (d) 2
16. 16.Which of these numbers is divisible by 4? (a) 1110 (b) 1103 (c) 1104 (d) 1105
17. 17.Which of these numbers is divisible by 6? (a) 1559 (b) 1554 (c) 1563 (d) 1552
18. 18.What is the units digit of 3^56? (a) 3 (b) 7 (c) 9 (d) 1
19. 19.What is the remainder when 286 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
20. 20.What is the remainder when 62 is divided by 6? (a) 2 (b) 1 (c) 3 (d) 4
21. 21.What is the units digit of 8^27? (a) 2 (b) 6 (c) 8 (d) 4
22. 22.What is the units digit of 3^43? (a) 1 (b) 3 (c) 9 (d) 7
23. 23.What is the remainder when 271 is divided by 12? (a) 7 (b) 8 (c) 6 (d) 5
24. 24.What is the smallest number you must add to 495 to make it divisible by 7? (a) 2 (b) 5 (c) 3 (d) 1
25. 25.Which of these numbers is divisible by 3? (a) 1623 (b) 1624 (c) 1627 (d) 1621
26. 26.What is the smallest number you must add to 110 to make it divisible by 8? (a) 3 (b) 2 (c) 6 (d) 1
27. 27.What is the units digit of 9^18? (a) 9 (b) 2 (c) 1 (d) 0
28. 28.Which of these numbers is divisible by 3? (a) 2011 (b) 2010 (c) 2009 (d) 2015
29. 29.What is the smallest number you must add to 594 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
30. 30.What is the remainder when 613 is divided by 11? (a) 8 (b) 3 (c) 7 (d) 9
31. 31.What is the units digit of 3^24? (a) 3 (b) 1 (c) 7 (d) 9
32. 32.What is the remainder when 943 is divided by 12? (a) 7 (b) 8 (c) 6 (d) 5
33. 33.Which of these numbers is divisible by 3? (a) 376 (b) 380 (c) 383 (d) 378
34. 34.Which of these numbers is divisible by 3? (a) 2662 (b) 2660 (c) 2661 (d) 2665
35. 35.What is the smallest number you must add to 181 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
36. 36.What is the remainder when 419 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 5
37. 37.What is the smallest number you must add to 114 to make it divisible by 8? (a) 7 (b) 5 (c) 2 (d) 6
38. 38.What is the smallest number you must add to 233 to make it divisible by 9? (a) 2 (b) 1 (c) 8 (d) 3
39. 39.What is the units digit of 8^42? (a) 6 (b) 8 (c) 4 (d) 2
40. 40.Which of these numbers is divisible by 11? (a) 3806 (b) 3825 (c) 3805 (d) 3816

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