Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 497 · 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 9? (a) 4974 (b) 4977 (c) 4994 (d) 4981
2. 2.What is the units digit of 2^25? (a) 4 (b) 2 (c) 6 (d) 8
3. 3.Which of these numbers is divisible by 11? (a) 6098 (b) 6107 (c) 6105 (d) 6120
4. 4.What is the remainder when 181 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
5. 5.What is the smallest number you must add to 619 to make it divisible by 10? (a) 2 (b) 3 (c) 9 (d) 1
6. 6.What is the remainder when 436 is divided by 10? (a) 6 (b) 5 (c) 4 (d) 7
7. 7.Which of these numbers is divisible by 4? (a) 2308 (b) 2315 (c) 2306 (d) 2309
8. 8.What is the units digit of 9^35? (a) 7 (b) 8 (c) 9 (d) 1
9. 9.Which of these numbers is divisible by 9? (a) 444 (b) 460 (c) 453 (d) 450
10. 10.Which of these numbers is divisible by 6? (a) 5124 (b) 5126 (c) 5135 (d) 5121
11. 11.What is the remainder when 327 is divided by 8? (a) 6 (b) 7 (c) 1 (d) 5
12. 12.What is the units digit of 4^33? (a) 4 (b) 5 (c) 6 (d) 3
13. 13.Which of these numbers is divisible by 6? (a) 624 (b) 628 (c) 635 (d) 619
14. 14.What is the units digit of 2^36? (a) 2 (b) 8 (c) 6 (d) 4
15. 15.What is the smallest number you must add to 708 to make it divisible by 9? (a) 3 (b) 6 (c) 2 (d) 4
16. 16.What is the remainder when 196 is divided by 3? (a) 2 (b) 1 (c) 0
17. 17.What is the smallest number you must add to 446 to make it divisible by 8? (a) 6 (b) 3 (c) 2 (d) 1
18. 18.What is the units digit of 2^47? (a) 2 (b) 6 (c) 8 (d) 4
19. 19.Which of these numbers is divisible by 4? (a) 1548 (b) 1549 (c) 1547 (d) 1554
20. 20.What is the smallest number you must add to 73 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
21. 21.What is the remainder when 221 is divided by 3? (a) 2 (b) 0 (c) 1
22. 22.Which of these numbers is divisible by 8? (a) 5486 (b) 5503 (c) 5488 (d) 5491
23. 23.What is the units digit of 2^18? (a) 8 (b) 4 (c) 6 (d) 2
24. 24.What is the smallest number you must add to 471 to make it divisible by 10? (a) 7 (b) 8 (c) 1 (d) 9
25. 25.Which of these numbers is divisible by 4? (a) 1023 (b) 1031 (c) 1027 (d) 1024
26. 26.What is the remainder when 887 is divided by 12? (a) 9 (b) 11 (c) 1 (d) 10
27. 27.What is the smallest number you must add to 885 to make it divisible by 11? (a) 6 (b) 8 (c) 5 (d) 7
28. 28.What is the smallest number you must add to 166 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
29. 29.What is the remainder when 140 is divided by 9? (a) 4 (b) 7 (c) 5 (d) 6
30. 30.What is the remainder when 129 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
31. 31.What is the units digit of 2^32? (a) 2 (b) 4 (c) 8 (d) 6
32. 32.What is the smallest number you must add to 493 to make it divisible by 12? (a) 11 (b) 1 (c) 10 (d) 9
33. 33.Which of these numbers is divisible by 4? (a) 707 (b) 703 (c) 700 (d) 697
34. 34.What is the remainder when 744 is divided by 10? (a) 3 (b) 6 (c) 5 (d) 4
35. 35.What is the units digit of 2^34? (a) 2 (b) 8 (c) 4 (d) 6
36. 36.What is the smallest number you must add to 506 to make it divisible by 12? (a) 10 (b) 2 (c) 9 (d) 11
37. 37.What is the units digit of 9^25? (a) 1 (b) 7 (c) 9 (d) 8
38. 38.Which of these numbers is divisible by 11? (a) 4081 (b) 4095 (c) 4087 (d) 4080
39. 39.Which of these numbers is divisible by 6? (a) 1313 (b) 1314 (c) 1321 (d) 1318
40. 40.What is the units digit of 7^50? (a) 7 (b) 3 (c) 1 (d) 9

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