Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 603 · 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 4^37? (a) 5 (b) 4 (c) 6 (d) 3
2. 2.Which of these numbers is divisible by 4? (a) 1631 (b) 1633 (c) 1627 (d) 1628
3. 3.What is the smallest number you must add to 249 to make it divisible by 4? (a) 1 (b) 2 (c) 3
4. 4.What is the remainder when 323 is divided by 9? (a) 6 (b) 8 (c) 1 (d) 7
5. 5.Which of these numbers is divisible by 9? (a) 2504 (b) 2495 (c) 2487 (d) 2493
6. 6.Which of these numbers is divisible by 3? (a) 131 (b) 128 (c) 126 (d) 125
7. 7.What is the units digit of 8^13? (a) 2 (b) 4 (c) 6 (d) 8
8. 8.What is the remainder when 388 is divided by 7? (a) 5 (b) 2 (c) 4 (d) 3
9. 9.What is the units digit of 2^13? (a) 4 (b) 6 (c) 8 (d) 2
10. 10.What is the smallest number you must add to 280 to make it divisible by 6? (a) 2 (b) 1 (c) 4 (d) 3
11. 11.What is the units digit of 2^24? (a) 4 (b) 8 (c) 6 (d) 2
12. 12.Which of these numbers is divisible by 3? (a) 615 (b) 616 (c) 614 (d) 619
13. 13.What is the smallest number you must add to 112 to make it divisible by 3? (a) 1 (b) 2
14. 14.Which of these numbers is divisible by 3? (a) 907 (b) 906 (c) 910 (d) 904
15. 15.What is the smallest number you must add to 94 to make it divisible by 3? (a) 1 (b) 2
16. 16.Which of these numbers is divisible by 8? (a) 3842 (b) 3836 (c) 3830 (d) 3832
17. 17.Which of these numbers is divisible by 6? (a) 1456 (b) 1446 (c) 1442 (d) 1450
18. 18.What is the units digit of 8^54? (a) 6 (b) 4 (c) 2 (d) 8
19. 19.What is the smallest number you must add to 566 to make it divisible by 10? (a) 6 (b) 3 (c) 5 (d) 4
20. 20.Which of these numbers is divisible by 9? (a) 6641 (b) 6643 (c) 6642 (d) 6657
21. 21.What is the units digit of 2^16? (a) 2 (b) 8 (c) 4 (d) 6
22. 22.What is the smallest number you must add to 253 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 5
23. 23.Which of these numbers is divisible by 3? (a) 1582 (b) 1578 (c) 1576 (d) 1579
24. 24.What is the units digit of 2^41? (a) 8 (b) 6 (c) 4 (d) 2
25. 25.What is the remainder when 413 is divided by 12? (a) 7 (b) 4 (c) 5 (d) 6
26. 26.Which of these numbers is divisible by 3? (a) 1497 (b) 1501 (c) 1496 (d) 1498
27. 27.What is the units digit of 7^11? (a) 3 (b) 9 (c) 1 (d) 7
28. 28.Which of these numbers is divisible by 4? (a) 628 (b) 631 (c) 626 (d) 635
29. 29.Which of these numbers is divisible by 11? (a) 3272 (b) 3288 (c) 3267 (d) 3263
30. 30.Which of these numbers is divisible by 9? (a) 7731 (b) 7743 (c) 7727 (d) 7737
31. 31.What is the smallest number you must add to 322 to make it divisible by 4? (a) 1 (b) 2 (c) 3
32. 32.What is the units digit of 8^25? (a) 8 (b) 4 (c) 2 (d) 6
33. 33.What is the remainder when 661 is divided by 12? (a) 11 (b) 2 (c) 1 (d) 0
34. 34.What is the units digit of 8^36? (a) 2 (b) 4 (c) 8 (d) 6
35. 35.What is the smallest number you must add to 449 to make it divisible by 10? (a) 3 (b) 2 (c) 9 (d) 1
36. 36.What is the smallest number you must add to 145 to make it divisible by 9? (a) 8 (b) 7 (c) 1 (d) 6
37. 37.What is the smallest number you must add to 226 to make it divisible by 8? (a) 2 (b) 6 (c) 7 (d) 5
38. 38.What is the units digit of 7^16? (a) 7 (b) 9 (c) 1 (d) 3
39. 39.What is the smallest number you must add to 414 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 1
40. 40.What is the remainder when 92 is divided by 6? (a) 3 (b) 1 (c) 4 (d) 2

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