Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 195 · 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 11? (a) 7624 (b) 7612 (c) 7610 (d) 7614
2. 2.Which of these numbers is divisible by 11? (a) 678 (b) 671 (c) 689 (d) 668
3. 3.What is the smallest number you must add to 417 to make it divisible by 10? (a) 3 (b) 7 (c) 4 (d) 2
4. 4.What is the units digit of 2^31? (a) 8 (b) 6 (c) 4 (d) 2
5. 5.Which of these numbers is divisible by 11? (a) 1912 (b) 1920 (c) 1903 (d) 1893
6. 6.What is the smallest number you must add to 437 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 5
7. 7.What is the units digit of 2^30? (a) 2 (b) 4 (c) 6 (d) 8
8. 8.What is the smallest number you must add to 290 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 2
9. 9.What is the smallest number you must add to 52 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
10. 10.What is the remainder when 42 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
11. 11.What is the units digit of 2^26? (a) 8 (b) 2 (c) 4 (d) 6
12. 12.Which of these numbers is divisible by 8? (a) 3100 (b) 3091 (c) 3096 (d) 3110
13. 13.What is the remainder when 95 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
14. 14.What is the remainder when 599 is divided by 10? (a) 8 (b) 7 (c) 9 (d) 1
15. 15.What is the remainder when 243 is divided by 11? (a) 10 (b) 2 (c) 1 (d) 0
16. 16.What is the remainder when 215 is divided by 6? (a) 3 (b) 1 (c) 5 (d) 4
17. 17.What is the remainder when 232 is divided by 11? (a) 2 (b) 0 (c) 10 (d) 1
18. 18.Which of these numbers is divisible by 9? (a) 7679 (b) 7691 (c) 7677 (d) 7674
19. 19.What is the units digit of 2^16? (a) 4 (b) 2 (c) 6 (d) 8
20. 20.Which of these numbers is divisible by 3? (a) 2561 (b) 2556 (c) 2557 (d) 2555
21. 21.What is the remainder when 638 is divided by 9? (a) 1 (b) 8 (c) 6 (d) 7
22. 22.What is the smallest number you must add to 156 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
23. 23.Which of these numbers is divisible by 6? (a) 5193 (b) 5188 (c) 5190 (d) 5198
24. 24.What is the smallest number you must add to 171 to make it divisible by 4? (a) 2 (b) 1 (c) 3
25. 25.Which of these numbers is divisible by 9? (a) 6659 (b) 6660 (c) 6676 (d) 6663
26. 26.What is the remainder when 67 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
27. 27.Which of these numbers is divisible by 8? (a) 454 (b) 448 (c) 446 (d) 460
28. 28.Which of these numbers is divisible by 4? (a) 2765 (b) 2760 (c) 2759 (d) 2763
29. 29.Which of these numbers is divisible by 9? (a) 6609 (b) 6597 (c) 6605 (d) 6591
30. 30.What is the units digit of 2^44? (a) 8 (b) 4 (c) 6 (d) 2
31. 31.What is the units digit of 3^42? (a) 9 (b) 1 (c) 3 (d) 7
32. 32.What is the units digit of 4^11? (a) 4 (b) 5 (c) 3 (d) 6
33. 33.What is the remainder when 167 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 1
34. 34.What is the remainder when 304 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 3
35. 35.What is the units digit of 7^20? (a) 3 (b) 7 (c) 1 (d) 9
36. 36.What is the units digit of 7^56? (a) 7 (b) 1 (c) 9 (d) 3
37. 37.What is the remainder when 223 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
38. 38.What is the units digit of 2^37? (a) 6 (b) 2 (c) 4 (d) 8
39. 39.Which of these numbers is divisible by 4? (a) 577 (b) 583 (c) 573 (d) 576
40. 40.What is the smallest number you must add to 629 to make it divisible by 12? (a) 7 (b) 5 (c) 8 (d) 6

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