Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 734 · 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 3? (a) 1709 (b) 1704 (c) 1705 (d) 1702
2. 2.What is the remainder when 223 is divided by 3? (a) 1 (b) 0 (c) 2
3. 3.What is the remainder when 200 is divided by 3? (a) 1 (b) 2 (c) 0
4. 4.What is the remainder when 719 is divided by 9? (a) 1 (b) 7 (c) 6 (d) 8
5. 5.Which of these numbers is divisible by 3? (a) 1550 (b) 1544 (c) 1545 (d) 1546
6. 6.What is the remainder when 217 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
7. 7.Which of these numbers is divisible by 6? (a) 4458 (b) 4460 (c) 4455 (d) 4468
8. 8.What is the units digit of 7^18? (a) 1 (b) 3 (c) 7 (d) 9
9. 9.What is the smallest number you must add to 729 to make it divisible by 10? (a) 1 (b) 9 (c) 2 (d) 3
10. 10.What is the units digit of 2^14? (a) 6 (b) 2 (c) 8 (d) 4
11. 11.What is the smallest number you must add to 769 to make it divisible by 12? (a) 1 (b) 9 (c) 11 (d) 10
12. 12.Which of these numbers is divisible by 11? (a) 589 (b) 565 (c) 582 (d) 572
13. 13.What is the remainder when 81 is divided by 8? (a) 0 (b) 1 (c) 2 (d) 7
14. 14.Which of these numbers is divisible by 3? (a) 225 (b) 227 (c) 229 (d) 223
15. 15.Which of these numbers is divisible by 11? (a) 6179 (b) 6171 (c) 6163 (d) 6189
16. 16.What is the smallest number you must add to 154 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
17. 17.What is the remainder when 461 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
18. 18.What is the remainder when 367 is divided by 12? (a) 6 (b) 7 (c) 5 (d) 8
19. 19.What is the remainder when 430 is divided by 9? (a) 6 (b) 2 (c) 8 (d) 7
20. 20.What is the remainder when 682 is divided by 10? (a) 8 (b) 2 (c) 1 (d) 3
21. 21.What is the units digit of 8^26? (a) 8 (b) 2 (c) 6 (d) 4
22. 22.What is the remainder when 231 is divided by 10? (a) 9 (b) 2 (c) 1 (d) 0
23. 23.What is the remainder when 205 is divided by 7? (a) 1 (b) 3 (c) 2 (d) 5
24. 24.What is the remainder when 243 is divided by 11? (a) 1 (b) 10 (c) 2 (d) 0
25. 25.Which of these numbers is divisible by 4? (a) 2706 (b) 2710 (c) 2704 (d) 2701
26. 26.What is the units digit of 4^59? (a) 6 (b) 5 (c) 4 (d) 3
27. 27.What is the units digit of 8^38? (a) 6 (b) 4 (c) 2 (d) 8
28. 28.What is the units digit of 7^20? (a) 7 (b) 3 (c) 9 (d) 1
29. 29.Which of these numbers is divisible by 6? (a) 2395 (b) 2403 (c) 2391 (d) 2394
30. 30.What is the remainder when 69 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
31. 31.What is the remainder when 199 is divided by 3? (a) 2 (b) 1 (c) 0
32. 32.Which of these numbers is divisible by 8? (a) 2961 (b) 2957 (c) 2975 (d) 2960
33. 33.What is the smallest number you must add to 534 to make it divisible by 9? (a) 3 (b) 7 (c) 6 (d) 5
34. 34.What is the smallest number you must add to 69 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
35. 35.Which of these numbers is divisible by 9? (a) 7276 (b) 7257 (c) 7265 (d) 7263
36. 36.Which of these numbers is divisible by 8? (a) 2230 (b) 2218 (c) 2216 (d) 2210
37. 37.What is the smallest number you must add to 399 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 7
38. 38.What is the units digit of 8^37? (a) 2 (b) 4 (c) 8 (d) 6
39. 39.What is the units digit of 7^27? (a) 7 (b) 1 (c) 3 (d) 9
40. 40.Which of these numbers is divisible by 11? (a) 2599 (b) 2610 (c) 2607 (d) 2619

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