Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 181 · 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 253 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
2. 2.What is the smallest number you must add to 404 to make it divisible by 12? (a) 4 (b) 8 (c) 3 (d) 5
3. 3.Which of these numbers is divisible by 8? (a) 4610 (b) 4617 (c) 4616 (d) 4630
4. 4.What is the units digit of 2^49? (a) 6 (b) 8 (c) 2 (d) 4
5. 5.Which of these numbers is divisible by 9? (a) 5505 (b) 5483 (c) 5497 (d) 5490
6. 6.What is the remainder when 629 is divided by 8? (a) 5 (b) 4 (c) 6 (d) 3
7. 7.What is the remainder when 99 is divided by 6? (a) 3 (b) 2 (c) 5 (d) 4
8. 8.Which of these numbers is divisible by 11? (a) 3873 (b) 3858 (c) 3867 (d) 3861
9. 9.What is the units digit of 4^42? (a) 4 (b) 5 (c) 7 (d) 6
10. 10.What is the smallest number you must add to 190 to make it divisible by 4? (a) 2 (b) 1 (c) 3
11. 11.What is the smallest number you must add to 247 to make it divisible by 4? (a) 3 (b) 2 (c) 1
12. 12.What is the remainder when 305 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 5
13. 13.Which of these numbers is divisible by 9? (a) 2575 (b) 2583 (c) 2594 (d) 2591
14. 14.What is the units digit of 7^17? (a) 1 (b) 9 (c) 7 (d) 3
15. 15.Which of these numbers is divisible by 6? (a) 2513 (b) 2499 (c) 2502 (d) 2503
16. 16.What is the units digit of 4^59? (a) 6 (b) 3 (c) 5 (d) 4
17. 17.Which of these numbers is divisible by 11? (a) 9902 (b) 9900 (c) 9914 (d) 9894
18. 18.What is the smallest number you must add to 790 to make it divisible by 12? (a) 3 (b) 1 (c) 10 (d) 2
19. 19.What is the remainder when 315 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
20. 20.What is the smallest number you must add to 83 to make it divisible by 8? (a) 6 (b) 3 (c) 5 (d) 4
21. 21.What is the smallest number you must add to 506 to make it divisible by 12? (a) 11 (b) 9 (c) 2 (d) 10
22. 22.What is the remainder when 523 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 5
23. 23.What is the units digit of 8^28? (a) 4 (b) 6 (c) 2 (d) 8
24. 24.What is the remainder when 187 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
25. 25.What is the remainder when 489 is divided by 11? (a) 7 (b) 4 (c) 5 (d) 6
26. 26.What is the remainder when 162 is divided by 7? (a) 0 (b) 6 (c) 2 (d) 1
27. 27.What is the units digit of 4^40? (a) 5 (b) 6 (c) 4 (d) 7
28. 28.Which of these numbers is divisible by 4? (a) 3285 (b) 3280 (c) 3279 (d) 3282
29. 29.What is the smallest number you must add to 294 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
30. 30.Which of these numbers is divisible by 9? (a) 6751 (b) 6737 (c) 6744 (d) 6741
31. 31.What is the units digit of 3^40? (a) 7 (b) 3 (c) 9 (d) 1
32. 32.What is the remainder when 271 is divided by 10? (a) 9 (b) 2 (c) 1 (d) 0
33. 33.Which of these numbers is divisible by 3? (a) 2651 (b) 2653 (c) 2649 (d) 2647
34. 34.What is the smallest number you must add to 314 to make it divisible by 10? (a) 4 (b) 5 (c) 6 (d) 7
35. 35.Which of these numbers is divisible by 8? (a) 1618 (b) 1614 (c) 1628 (d) 1616
36. 36.Which of these numbers is divisible by 9? (a) 2809 (b) 2804 (c) 2820 (d) 2808
37. 37.What is the units digit of 4^26? (a) 7 (b) 4 (c) 6 (d) 5
38. 38.What is the units digit of 4^47? (a) 6 (b) 5 (c) 4 (d) 3
39. 39.Which of these numbers is divisible by 6? (a) 1759 (b) 1774 (c) 1769 (d) 1764
40. 40.What is the smallest number you must add to 155 to make it divisible by 4? (a) 2 (b) 1 (c) 3

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