Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 1 · 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 4? (a) 997 (b) 1000 (c) 1001 (d) 1005
2. 2.Which of these numbers is divisible by 11? (a) 7774 (b) 7764 (c) 7755 (d) 7750
3. 3.What is the smallest number you must add to 49 to make it divisible by 3? (a) 1 (b) 2
4. 4.Which of these numbers is divisible by 9? (a) 2431 (b) 2422 (c) 2430 (d) 2445
5. 5.Which of these numbers is divisible by 3? (a) 334 (b) 333 (c) 331 (d) 338
6. 6.What is the remainder when 83 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
7. 7.Which of these numbers is divisible by 4? (a) 1235 (b) 1227 (c) 1229 (d) 1228
8. 8.Which of these numbers is divisible by 9? (a) 3011 (b) 3006 (c) 3023 (d) 2999
9. 9.Which of these numbers is divisible by 8? (a) 6873 (b) 6872 (c) 6884 (d) 6868
10. 10.Which of these numbers is divisible by 6? (a) 4503 (b) 4510 (c) 4498 (d) 4500
11. 11.What is the remainder when 649 is divided by 9? (a) 0 (b) 8 (c) 2 (d) 1
12. 12.What is the units digit of 3^59? (a) 1 (b) 7 (c) 9 (d) 3
13. 13.What is the smallest number you must add to 193 to make it divisible by 9? (a) 7 (b) 6 (c) 4 (d) 5
14. 14.What is the units digit of 8^27? (a) 8 (b) 4 (c) 2 (d) 6
15. 15.What is the units digit of 4^15? (a) 3 (b) 4 (c) 5 (d) 6
16. 16.What is the remainder when 286 is divided by 7? (a) 4 (b) 6 (c) 1 (d) 5
17. 17.What is the smallest number you must add to 122 to make it divisible by 4? (a) 1 (b) 3 (c) 2
18. 18.Which of these numbers is divisible by 8? (a) 6846 (b) 6837 (c) 6828 (d) 6832
19. 19.Which of these numbers is divisible by 4? (a) 2484 (b) 2489 (c) 2485 (d) 2482
20. 20.What is the smallest number you must add to 221 to make it divisible by 3? (a) 2 (b) 1
21. 21.Which of these numbers is divisible by 4? (a) 2820 (b) 2823 (c) 2817 (d) 2825
22. 22.Which of these numbers is divisible by 8? (a) 6112 (b) 6106 (c) 6118 (d) 6122
23. 23.What is the units digit of 3^30? (a) 1 (b) 9 (c) 3 (d) 7
24. 24.Which of these numbers is divisible by 9? (a) 2733 (b) 2727 (c) 2738 (d) 2726
25. 25.Which of these numbers is divisible by 8? (a) 2190 (b) 2193 (c) 2179 (d) 2184
26. 26.What is the units digit of 3^32? (a) 7 (b) 1 (c) 3 (d) 9
27. 27.What is the remainder when 286 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4
28. 28.What is the units digit of 3^21? (a) 7 (b) 3 (c) 1 (d) 9
29. 29.What is the smallest number you must add to 291 to make it divisible by 4? (a) 1 (b) 3 (c) 2
30. 30.Which of these numbers is divisible by 6? (a) 2410 (b) 2420 (c) 2417 (d) 2412
31. 31.Which of these numbers is divisible by 11? (a) 2079 (b) 2095 (c) 2081 (d) 2070
32. 32.What is the smallest number you must add to 691 to make it divisible by 10? (a) 1 (b) 8 (c) 7 (d) 9
33. 33.What is the smallest number you must add to 119 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
34. 34.What is the remainder when 167 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
35. 35.What is the units digit of 4^25? (a) 4 (b) 5 (c) 6 (d) 3
36. 36.What is the units digit of 9^59? (a) 7 (b) 1 (c) 8 (d) 9
37. 37.What is the smallest number you must add to 52 to make it divisible by 3? (a) 2 (b) 1
38. 38.What is the smallest number you must add to 206 to make it divisible by 3? (a) 1 (b) 2
39. 39.What is the remainder when 186 is divided by 11? (a) 10 (b) 8 (c) 9 (d) 1
40. 40.Which of these numbers is divisible by 4? (a) 589 (b) 584 (c) 585 (d) 582

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