Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 588 · 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 units digit of 8^24? (a) 8 (b) 2 (c) 4 (d) 6
2. 2.What is the remainder when 696 is divided by 11? (a) 4 (b) 3 (c) 2 (d) 8
3. 3.What is the smallest number you must add to 641 to make it divisible by 9? (a) 8 (b) 6 (c) 7 (d) 2
4. 4.What is the units digit of 9^59? (a) 1 (b) 8 (c) 7 (d) 9
5. 5.What is the smallest number you must add to 380 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
6. 6.What is the units digit of 2^29? (a) 8 (b) 2 (c) 6 (d) 4
7. 7.Which of these numbers is divisible by 9? (a) 7104 (b) 7110 (c) 7118 (d) 7126
8. 8.What is the smallest number you must add to 191 to make it divisible by 3? (a) 2 (b) 1
9. 9.Which of these numbers is divisible by 6? (a) 381 (b) 378 (c) 385 (d) 376
10. 10.Which of these numbers is divisible by 6? (a) 2567 (b) 2557 (c) 2551 (d) 2556
11. 11.What is the remainder when 121 is divided by 10? (a) 0 (b) 9 (c) 1 (d) 2
12. 12.Which of these numbers is divisible by 11? (a) 1328 (b) 1351 (c) 1336 (d) 1331
13. 13.What is the remainder when 469 is divided by 12? (a) 2 (b) 11 (c) 0 (d) 1
14. 14.What is the units digit of 2^17? (a) 8 (b) 2 (c) 6 (d) 4
15. 15.What is the smallest number you must add to 592 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
16. 16.What is the units digit of 9^17? (a) 9 (b) 1 (c) 7 (d) 8
17. 17.Which of these numbers is divisible by 4? (a) 2693 (b) 2689 (c) 2697 (d) 2692
18. 18.Which of these numbers is divisible by 11? (a) 7752 (b) 7762 (c) 7738 (d) 7744
19. 19.What is the units digit of 2^60? (a) 4 (b) 2 (c) 6 (d) 8
20. 20.What is the units digit of 3^23? (a) 3 (b) 1 (c) 7 (d) 9
21. 21.Which of these numbers is divisible by 11? (a) 8778 (b) 8794 (c) 8772 (d) 8783
22. 22.What is the smallest number you must add to 169 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
23. 23.What is the units digit of 4^31? (a) 3 (b) 4 (c) 6 (d) 5
24. 24.What is the units digit of 9^25? (a) 8 (b) 7 (c) 9 (d) 1
25. 25.What is the units digit of 8^36? (a) 6 (b) 4 (c) 2 (d) 8
26. 26.What is the units digit of 3^21? (a) 7 (b) 9 (c) 3 (d) 1
27. 27.Which of these numbers is divisible by 11? (a) 1082 (b) 1062 (c) 1067 (d) 1071
28. 28.What is the remainder when 56 is divided by 3? (a) 2 (b) 0 (c) 1
29. 29.What is the smallest number you must add to 44 to make it divisible by 3? (a) 1 (b) 2
30. 30.What is the smallest number you must add to 304 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
31. 31.What is the smallest number you must add to 141 to make it divisible by 4? (a) 1 (b) 2 (c) 3
32. 32.What is the remainder when 93 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 1
33. 33.What is the remainder when 41 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
34. 34.What is the remainder when 473 is divided by 10? (a) 2 (b) 4 (c) 3 (d) 7
35. 35.What is the remainder when 485 is divided by 8? (a) 5 (b) 3 (c) 6 (d) 4
36. 36.What is the smallest number you must add to 146 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
37. 37.Which of these numbers is divisible by 8? (a) 2010 (b) 2027 (c) 2019 (d) 2016
38. 38.Which of these numbers is divisible by 4? (a) 645 (b) 649 (c) 644 (d) 641
39. 39.What is the units digit of 8^18? (a) 2 (b) 8 (c) 6 (d) 4
40. 40.Which of these numbers is divisible by 11? (a) 7633 (b) 7651 (c) 7634 (d) 7639

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