Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 984 · 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 8? (a) 3281 (b) 3278 (c) 3272 (d) 3267
2. 2.What is the units digit of 9^44? (a) 9 (b) 0 (c) 1 (d) 2
3. 3.Which of these numbers is divisible by 8? (a) 5660 (b) 5656 (c) 5667 (d) 5655
4. 4.What is the smallest number you must add to 292 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
5. 5.Which of these numbers is divisible by 8? (a) 3591 (b) 3592 (c) 3599 (d) 3606
6. 6.What is the smallest number you must add to 783 to make it divisible by 11? (a) 8 (b) 10 (c) 2 (d) 9
7. 7.Which of these numbers is divisible by 4? (a) 1597 (b) 1594 (c) 1596 (d) 1601
8. 8.Which of these numbers is divisible by 4? (a) 1685 (b) 1682 (c) 1691 (d) 1684
9. 9.Which of these numbers is divisible by 11? (a) 8206 (b) 8222 (c) 8199 (d) 8215
10. 10.What is the smallest number you must add to 469 to make it divisible by 11? (a) 3 (b) 7 (c) 5 (d) 4
11. 11.Which of these numbers is divisible by 4? (a) 2590 (b) 2587 (c) 2582 (d) 2584
12. 12.What is the remainder when 226 is divided by 7? (a) 5 (b) 2 (c) 3 (d) 1
13. 13.What is the smallest number you must add to 581 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 2
14. 14.Which of these numbers is divisible by 9? (a) 2828 (b) 2823 (c) 2840 (d) 2826
15. 15.What is the units digit of 4^50? (a) 6 (b) 4 (c) 5 (d) 7
16. 16.What is the remainder when 731 is divided by 11? (a) 7 (b) 4 (c) 5 (d) 6
17. 17.What is the remainder when 231 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
18. 18.What is the remainder when 892 is divided by 12? (a) 5 (b) 3 (c) 4 (d) 8
19. 19.What is the remainder when 97 is divided by 3? (a) 0 (b) 1 (c) 2
20. 20.Which of these numbers is divisible by 4? (a) 2774 (b) 2777 (c) 2772 (d) 2769
21. 21.What is the units digit of 3^47? (a) 9 (b) 1 (c) 3 (d) 7
22. 22.Which of these numbers is divisible by 6? (a) 2385 (b) 2378 (c) 2382 (d) 2393
23. 23.What is the smallest number you must add to 237 to make it divisible by 8? (a) 5 (b) 4 (c) 2 (d) 3
24. 24.What is the remainder when 127 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
25. 25.What is the smallest number you must add to 215 to make it divisible by 9? (a) 2 (b) 1 (c) 8 (d) 3
26. 26.What is the units digit of 7^20? (a) 7 (b) 3 (c) 1 (d) 9
27. 27.Which of these numbers is divisible by 4? (a) 2662 (b) 2666 (c) 2664 (d) 2669
28. 28.What is the remainder when 392 is divided by 9? (a) 5 (b) 7 (c) 4 (d) 6
29. 29.What is the remainder when 650 is divided by 9? (a) 7 (b) 1 (c) 2 (d) 3
30. 30.What is the units digit of 2^37? (a) 6 (b) 4 (c) 2 (d) 8
31. 31.What is the units digit of 4^37? (a) 4 (b) 3 (c) 6 (d) 5
32. 32.What is the remainder when 249 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
33. 33.Which of these numbers is divisible by 11? (a) 1584 (b) 1577 (c) 1590 (d) 1605
34. 34.What is the remainder when 489 is divided by 8? (a) 2 (b) 7 (c) 1 (d) 0
35. 35.What is the smallest number you must add to 229 to make it divisible by 3? (a) 1 (b) 2
36. 36.What is the smallest number you must add to 247 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 6
37. 37.What is the units digit of 7^37? (a) 7 (b) 9 (c) 3 (d) 1
38. 38.What is the smallest number you must add to 206 to make it divisible by 7? (a) 5 (b) 3 (c) 6 (d) 4
39. 39.What is the remainder when 422 is divided by 7? (a) 5 (b) 3 (c) 2 (d) 1
40. 40.What is the units digit of 9^35? (a) 9 (b) 7 (c) 1 (d) 8

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