Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 183 · 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 675 is divided by 11? (a) 3 (b) 7 (c) 5 (d) 4
2. 2.Which of these numbers is divisible by 11? (a) 1200 (b) 1216 (c) 1224 (d) 1210
3. 3.What is the units digit of 4^39? (a) 3 (b) 6 (c) 4 (d) 5
4. 4.What is the smallest number you must add to 55 to make it divisible by 4? (a) 3 (b) 1 (c) 2
5. 5.Which of these numbers is divisible by 4? (a) 3307 (b) 3308 (c) 3310 (d) 3314
6. 6.Which of these numbers is divisible by 4? (a) 2868 (b) 2865 (c) 2875 (d) 2869
7. 7.What is the units digit of 8^60? (a) 2 (b) 4 (c) 6 (d) 8
8. 8.What is the units digit of 2^49? (a) 2 (b) 4 (c) 6 (d) 8
9. 9.What is the remainder when 809 is divided by 12? (a) 6 (b) 5 (c) 4 (d) 7
10. 10.Which of these numbers is divisible by 4? (a) 883 (b) 887 (c) 884 (d) 891
11. 11.What is the smallest number you must add to 204 to make it divisible by 7? (a) 4 (b) 1 (c) 6 (d) 5
12. 12.What is the smallest number you must add to 118 to make it divisible by 11? (a) 8 (b) 3 (c) 4 (d) 2
13. 13.What is the smallest number you must add to 450 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 2
14. 14.Which of these numbers is divisible by 4? (a) 2621 (b) 2616 (c) 2617 (d) 2615
15. 15.What is the smallest number you must add to 506 to make it divisible by 7? (a) 4 (b) 6 (c) 2 (d) 5
16. 16.Which of these numbers is divisible by 4? (a) 3255 (b) 3257 (c) 3256 (d) 3263
17. 17.Which of these numbers is divisible by 11? (a) 8365 (b) 8360 (c) 8372 (d) 8355
18. 18.What is the smallest number you must add to 49 to make it divisible by 4? (a) 3 (b) 1 (c) 2
19. 19.What is the remainder when 593 is divided by 8? (a) 1 (b) 2 (c) 7 (d) 0
20. 20.Which of these numbers is divisible by 3? (a) 416 (b) 412 (c) 418 (d) 414
21. 21.What is the smallest number you must add to 346 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
22. 22.What is the units digit of 7^16? (a) 7 (b) 3 (c) 1 (d) 9
23. 23.What is the units digit of 9^47? (a) 9 (b) 1 (c) 7 (d) 8
24. 24.What is the units digit of 2^47? (a) 6 (b) 4 (c) 2 (d) 8
25. 25.Which of these numbers is divisible by 6? (a) 4519 (b) 4533 (c) 4528 (d) 4524
26. 26.What is the remainder when 519 is divided by 8? (a) 6 (b) 7 (c) 1 (d) 5
27. 27.Which of these numbers is divisible by 6? (a) 3804 (b) 3815 (c) 3806 (d) 3803
28. 28.What is the units digit of 9^16? (a) 2 (b) 1 (c) 9 (d) 0
29. 29.What is the remainder when 258 is divided by 8? (a) 3 (b) 2 (c) 1 (d) 6
30. 30.Which of these numbers is divisible by 3? (a) 1590 (b) 1595 (c) 1589 (d) 1591
31. 31.What is the remainder when 442 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
32. 32.What is the smallest number you must add to 115 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 1
33. 33.What is the units digit of 9^10? (a) 1 (b) 0 (c) 2 (d) 9
34. 34.What is the smallest number you must add to 580 to make it divisible by 11? (a) 8 (b) 2 (c) 4 (d) 3
35. 35.What is the remainder when 160 is divided by 6? (a) 3 (b) 5 (c) 2 (d) 4
36. 36.Which of these numbers is divisible by 3? (a) 873 (b) 872 (c) 878 (d) 875
37. 37.What is the remainder when 100 is divided by 3? (a) 0 (b) 1 (c) 2
38. 38.What is the remainder when 337 is divided by 9? (a) 4 (b) 5 (c) 3 (d) 6
39. 39.What is the remainder when 133 is divided by 11? (a) 0 (b) 1 (c) 10 (d) 2
40. 40.What is the smallest number you must add to 75 to make it divisible by 4? (a) 1 (b) 3 (c) 2

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