Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 323 · 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 371 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
2. 2.What is the units digit of 8^50? (a) 6 (b) 2 (c) 4 (d) 8
3. 3.What is the smallest number you must add to 736 to make it divisible by 11? (a) 2 (b) 10 (c) 3 (d) 1
4. 4.What is the units digit of 9^37? (a) 8 (b) 1 (c) 9 (d) 7
5. 5.What is the remainder when 554 is divided by 11? (a) 5 (b) 7 (c) 4 (d) 3
6. 6.Which of these numbers is divisible by 6? (a) 710 (b) 717 (c) 721 (d) 714
7. 7.What is the units digit of 7^24? (a) 1 (b) 9 (c) 3 (d) 7
8. 8.What is the remainder when 100 is divided by 8? (a) 6 (b) 5 (c) 4 (d) 3
9. 9.Which of these numbers is divisible by 6? (a) 2779 (b) 2773 (c) 2789 (d) 2778
10. 10.What is the remainder when 66 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
11. 11.What is the smallest number you must add to 181 to make it divisible by 3? (a) 2 (b) 1
12. 12.What is the remainder when 160 is divided by 7? (a) 4 (b) 5 (c) 1 (d) 6
13. 13.What is the smallest number you must add to 58 to make it divisible by 3? (a) 2 (b) 1
14. 14.Which of these numbers is divisible by 3? (a) 577 (b) 572 (c) 574 (d) 573
15. 15.Which of these numbers is divisible by 4? (a) 2500 (b) 2505 (c) 2498 (d) 2501
16. 16.What is the smallest number you must add to 276 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
17. 17.Which of these numbers is divisible by 8? (a) 3228 (b) 3216 (c) 3213 (d) 3222
18. 18.What is the smallest number you must add to 285 to make it divisible by 12? (a) 4 (b) 3 (c) 9 (d) 2
19. 19.What is the units digit of 4^38? (a) 5 (b) 6 (c) 7 (d) 4
20. 20.What is the remainder when 222 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
21. 21.What is the smallest number you must add to 94 to make it divisible by 4? (a) 3 (b) 1 (c) 2
22. 22.Which of these numbers is divisible by 3? (a) 1540 (b) 1542 (c) 1546 (d) 1543
23. 23.What is the smallest number you must add to 169 to make it divisible by 4? (a) 1 (b) 3 (c) 2
24. 24.What is the units digit of 4^13? (a) 4 (b) 6 (c) 3 (d) 5
25. 25.What is the units digit of 8^59? (a) 6 (b) 4 (c) 8 (d) 2
26. 26.What is the units digit of 9^50? (a) 0 (b) 9 (c) 1 (d) 2
27. 27.Which of these numbers is divisible by 9? (a) 7444 (b) 7456 (c) 7438 (d) 7443
28. 28.What is the smallest number you must add to 477 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
29. 29.Which of these numbers is divisible by 9? (a) 7409 (b) 7431 (c) 7416 (d) 7421
30. 30.What is the remainder when 529 is divided by 10? (a) 7 (b) 8 (c) 1 (d) 9
31. 31.What is the smallest number you must add to 447 to make it divisible by 7? (a) 3 (b) 6 (c) 1 (d) 2
32. 32.Which of these numbers is divisible by 4? (a) 2707 (b) 2714 (c) 2710 (d) 2708
33. 33.Which of these numbers is divisible by 9? (a) 1694 (b) 1701 (c) 1708 (d) 1711
34. 34.Which of these numbers is divisible by 4? (a) 1544 (b) 1546 (c) 1549 (d) 1541
35. 35.Which of these numbers is divisible by 4? (a) 2073 (b) 2078 (c) 2072 (d) 2071
36. 36.What is the smallest number you must add to 703 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 1
37. 37.What is the units digit of 2^16? (a) 6 (b) 2 (c) 8 (d) 4
38. 38.What is the units digit of 9^53? (a) 8 (b) 9 (c) 7 (d) 1
39. 39.What is the units digit of 7^50? (a) 3 (b) 1 (c) 9 (d) 7
40. 40.What is the smallest number you must add to 130 to make it divisible by 3? (a) 2 (b) 1

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