Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 474 · 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) 1821 (b) 1823 (c) 1825 (d) 1819
2. 2.What is the units digit of 9^28? (a) 2 (b) 0 (c) 1 (d) 9
3. 3.Which of these numbers is divisible by 3? (a) 725 (b) 726 (c) 728 (d) 730
4. 4.What is the remainder when 307 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
5. 5.What is the units digit of 2^40? (a) 4 (b) 8 (c) 6 (d) 2
6. 6.Which of these numbers is divisible by 9? (a) 992 (b) 987 (c) 976 (d) 981
7. 7.What is the remainder when 275 is divided by 8? (a) 3 (b) 2 (c) 4 (d) 5
8. 8.What is the remainder when 521 is divided by 12? (a) 7 (b) 4 (c) 6 (d) 5
9. 9.What is the smallest number you must add to 61 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
10. 10.What is the units digit of 8^24? (a) 2 (b) 4 (c) 8 (d) 6
11. 11.What is the remainder when 185 is divided by 3? (a) 1 (b) 0 (c) 2
12. 12.Which of these numbers is divisible by 8? (a) 3077 (b) 3066 (c) 3064 (d) 3057
13. 13.What is the remainder when 567 is divided by 12? (a) 4 (b) 9 (c) 2 (d) 3
14. 14.What is the smallest number you must add to 194 to make it divisible by 4? (a) 3 (b) 1 (c) 2
15. 15.What is the units digit of 8^28? (a) 8 (b) 2 (c) 4 (d) 6
16. 16.What is the smallest number you must add to 225 to make it divisible by 4? (a) 3 (b) 1 (c) 2
17. 17.What is the smallest number you must add to 138 to make it divisible by 12? (a) 8 (b) 5 (c) 6 (d) 7
18. 18.What is the remainder when 463 is divided by 6? (a) 1 (b) 0 (c) 2 (d) 5
19. 19.Which of these numbers is divisible by 11? (a) 5995 (b) 5996 (c) 5992 (d) 6013
20. 20.What is the units digit of 3^24? (a) 7 (b) 9 (c) 1 (d) 3
21. 21.What is the smallest number you must add to 70 to make it divisible by 3? (a) 2 (b) 1
22. 22.What is the smallest number you must add to 859 to make it divisible by 11? (a) 9 (b) 10 (c) 8 (d) 1
23. 23.What is the smallest number you must add to 152 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
24. 24.What is the remainder when 213 is divided by 9? (a) 7 (b) 5 (c) 3 (d) 6
25. 25.What is the smallest number you must add to 146 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 6
26. 26.What is the units digit of 8^19? (a) 6 (b) 4 (c) 2 (d) 8
27. 27.What is the remainder when 214 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
28. 28.Which of these numbers is divisible by 6? (a) 3348 (b) 3343 (c) 3352 (d) 3357
29. 29.Which of these numbers is divisible by 8? (a) 4860 (b) 4856 (c) 4866 (d) 4854
30. 30.Which of these numbers is divisible by 3? (a) 1206 (b) 1207 (c) 1204 (d) 1210
31. 31.What is the units digit of 2^47? (a) 2 (b) 8 (c) 6 (d) 4
32. 32.What is the remainder when 556 is divided by 9? (a) 6 (b) 7 (c) 8 (d) 2
33. 33.What is the units digit of 8^38? (a) 8 (b) 4 (c) 6 (d) 2
34. 34.What is the units digit of 2^16? (a) 8 (b) 4 (c) 6 (d) 2
35. 35.Which of these numbers is divisible by 3? (a) 1439 (b) 1434 (c) 1432 (d) 1436
36. 36.Which of these numbers is divisible by 6? (a) 4717 (b) 4710 (c) 4707 (d) 4714
37. 37.What is the smallest number you must add to 61 to make it divisible by 3? (a) 1 (b) 2
38. 38.What is the smallest number you must add to 66 to make it divisible by 4? (a) 1 (b) 2 (c) 3
39. 39.Which of these numbers is divisible by 3? (a) 2655 (b) 2656 (c) 2654 (d) 2659
40. 40.Which of these numbers is divisible by 9? (a) 565 (b) 549 (c) 545 (d) 556

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