Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 607 · 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) 1597 (b) 1599 (c) 1603 (d) 1601
2. 2.What is the units digit of 9^21? (a) 7 (b) 1 (c) 9 (d) 8
3. 3.What is the units digit of 4^20? (a) 6 (b) 5 (c) 7 (d) 4
4. 4.What is the smallest number you must add to 226 to make it divisible by 3? (a) 2 (b) 1
5. 5.Which of these numbers is divisible by 4? (a) 865 (b) 870 (c) 863 (d) 864
6. 6.What is the units digit of 3^19? (a) 9 (b) 3 (c) 7 (d) 1
7. 7.What is the remainder when 481 is divided by 11? (a) 9 (b) 7 (c) 8 (d) 3
8. 8.What is the remainder when 128 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
9. 9.What is the smallest number you must add to 200 to make it divisible by 9? (a) 2 (b) 8 (c) 7 (d) 6
10. 10.Which of these numbers is divisible by 6? (a) 2135 (b) 2139 (c) 2136 (d) 2144
11. 11.What is the smallest number you must add to 239 to make it divisible by 4? (a) 2 (b) 1 (c) 3
12. 12.What is the remainder when 217 is divided by 8? (a) 2 (b) 7 (c) 0 (d) 1
13. 13.Which of these numbers is divisible by 6? (a) 4524 (b) 4523 (c) 4533 (d) 4526
14. 14.What is the smallest number you must add to 421 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 2
15. 15.What is the units digit of 4^14? (a) 4 (b) 6 (c) 7 (d) 5
16. 16.What is the smallest number you must add to 317 to make it divisible by 8? (a) 5 (b) 4 (c) 2 (d) 3
17. 17.What is the remainder when 103 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
18. 18.What is the remainder when 318 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
19. 19.Which of these numbers is divisible by 6? (a) 1929 (b) 1917 (c) 1920 (d) 1922
20. 20.What is the units digit of 2^35? (a) 2 (b) 8 (c) 4 (d) 6
21. 21.What is the remainder when 266 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
22. 22.Which of these numbers is divisible by 4? (a) 495 (b) 491 (c) 488 (d) 485
23. 23.What is the units digit of 4^16? (a) 5 (b) 6 (c) 4 (d) 7
24. 24.What is the units digit of 7^38? (a) 1 (b) 9 (c) 7 (d) 3
25. 25.What is the remainder when 271 is divided by 10? (a) 2 (b) 1 (c) 9 (d) 0
26. 26.Which of these numbers is divisible by 8? (a) 824 (b) 829 (c) 823 (d) 837
27. 27.What is the units digit of 4^57? (a) 3 (b) 6 (c) 5 (d) 4
28. 28.What is the units digit of 9^19? (a) 1 (b) 7 (c) 9 (d) 8
29. 29.What is the smallest number you must add to 479 to make it divisible by 6? (a) 2 (b) 1 (c) 5 (d) 3
30. 30.What is the units digit of 7^19? (a) 7 (b) 1 (c) 9 (d) 3
31. 31.What is the units digit of 3^12? (a) 3 (b) 9 (c) 7 (d) 1
32. 32.What is the remainder when 442 is divided by 9? (a) 1 (b) 0 (c) 2 (d) 8
33. 33.What is the units digit of 9^31? (a) 9 (b) 7 (c) 8 (d) 1
34. 34.What is the units digit of 7^45? (a) 1 (b) 9 (c) 7 (d) 3
35. 35.What is the remainder when 258 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
36. 36.What is the remainder when 287 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 1
37. 37.What is the units digit of 9^16? (a) 2 (b) 0 (c) 1 (d) 9
38. 38.What is the units digit of 4^12? (a) 4 (b) 7 (c) 5 (d) 6
39. 39.Which of these numbers is divisible by 8? (a) 6393 (b) 6400 (c) 6409 (d) 6407
40. 40.What is the remainder when 245 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3

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