Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 888 · 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 smallest number you must add to 410 to make it divisible by 8? (a) 2 (b) 7 (c) 5 (d) 6
2. 2.What is the units digit of 3^20? (a) 1 (b) 9 (c) 3 (d) 7
3. 3.What is the smallest number you must add to 202 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
4. 4.What is the smallest number you must add to 263 to make it divisible by 7? (a) 5 (b) 2 (c) 4 (d) 3
5. 5.What is the remainder when 185 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
6. 6.What is the units digit of 9^39? (a) 1 (b) 7 (c) 9 (d) 8
7. 7.What is the remainder when 147 is divided by 11? (a) 4 (b) 5 (c) 7 (d) 3
8. 8.What is the remainder when 467 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 2
9. 9.What is the remainder when 342 is divided by 10? (a) 8 (b) 3 (c) 2 (d) 1
10. 10.What is the remainder when 895 is divided by 12? (a) 6 (b) 5 (c) 7 (d) 8
11. 11.Which of these numbers is divisible by 9? (a) 1045 (b) 1053 (c) 1063 (d) 1058
12. 12.What is the units digit of 4^26? (a) 5 (b) 7 (c) 6 (d) 4
13. 13.What is the remainder when 177 is divided by 8? (a) 2 (b) 0 (c) 1 (d) 7
14. 14.What is the smallest number you must add to 148 to make it divisible by 3? (a) 1 (b) 2
15. 15.What is the smallest number you must add to 154 to make it divisible by 9? (a) 1 (b) 7 (c) 6 (d) 8
16. 16.What is the remainder when 372 is divided by 9? (a) 4 (b) 3 (c) 6 (d) 2
17. 17.Which of these numbers is divisible by 8? (a) 547 (b) 544 (c) 553 (d) 542
18. 18.What is the remainder when 286 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 1
19. 19.Which of these numbers is divisible by 8? (a) 2925 (b) 2912 (c) 2911 (d) 2915
20. 20.What is the smallest number you must add to 500 to make it divisible by 7? (a) 3 (b) 6 (c) 5 (d) 4
21. 21.Which of these numbers is divisible by 4? (a) 3286 (b) 3290 (c) 3284 (d) 3281
22. 22.What is the remainder when 300 is divided by 8? (a) 3 (b) 4 (c) 5 (d) 6
23. 23.What is the units digit of 9^21? (a) 8 (b) 9 (c) 1 (d) 7
24. 24.What is the units digit of 4^14? (a) 5 (b) 4 (c) 7 (d) 6
25. 25.Which of these numbers is divisible by 8? (a) 6369 (b) 6360 (c) 6359 (d) 6365
26. 26.What is the units digit of 8^49? (a) 4 (b) 2 (c) 8 (d) 6
27. 27.Which of these numbers is divisible by 6? (a) 4944 (b) 4954 (c) 4941 (d) 4949
28. 28.What is the smallest number you must add to 212 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
29. 29.What is the remainder when 133 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
30. 30.What is the smallest number you must add to 198 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
31. 31.What is the units digit of 3^38? (a) 7 (b) 3 (c) 1 (d) 9
32. 32.Which of these numbers is divisible by 3? (a) 2092 (b) 2087 (c) 2088 (d) 2090
33. 33.Which of these numbers is divisible by 3? (a) 817 (b) 824 (c) 821 (d) 819
34. 34.What is the units digit of 7^12? (a) 3 (b) 1 (c) 9 (d) 7
35. 35.What is the units digit of 3^21? (a) 1 (b) 9 (c) 3 (d) 7
36. 36.Which of these numbers is divisible by 6? (a) 1583 (b) 1589 (c) 1584 (d) 1594
37. 37.Which of these numbers is divisible by 11? (a) 865 (b) 847 (c) 842 (d) 856
38. 38.What is the smallest number you must add to 581 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 9
39. 39.What is the units digit of 8^12? (a) 2 (b) 8 (c) 4 (d) 6
40. 40.What is the remainder when 329 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 1

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