Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 960 · 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) 1045 (b) 1039 (c) 1040 (d) 1042
2. 2.What is the smallest number you must add to 379 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 1
3. 3.What is the remainder when 540 is divided by 7? (a) 6 (b) 0 (c) 2 (d) 1
4. 4.Which of these numbers is divisible by 9? (a) 8091 (b) 8103 (c) 8093 (d) 8086
5. 5.What is the units digit of 2^30? (a) 6 (b) 2 (c) 8 (d) 4
6. 6.What is the units digit of 9^49? (a) 1 (b) 8 (c) 7 (d) 9
7. 7.What is the units digit of 3^27? (a) 9 (b) 1 (c) 3 (d) 7
8. 8.Which of these numbers is divisible by 9? (a) 3214 (b) 3222 (c) 3233 (d) 3225
9. 9.Which of these numbers is divisible by 6? (a) 4905 (b) 4900 (c) 4902 (d) 4909
10. 10.What is the units digit of 7^46? (a) 7 (b) 3 (c) 1 (d) 9
11. 11.Which of these numbers is divisible by 4? (a) 2815 (b) 2822 (c) 2816 (d) 2817
12. 12.What is the smallest number you must add to 811 to make it divisible by 12? (a) 6 (b) 7 (c) 5 (d) 4
13. 13.What is the units digit of 4^11? (a) 6 (b) 4 (c) 5 (d) 3
14. 14.What is the smallest number you must add to 128 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
15. 15.What is the units digit of 7^58? (a) 1 (b) 3 (c) 7 (d) 9
16. 16.What is the units digit of 3^22? (a) 3 (b) 7 (c) 1 (d) 9
17. 17.Which of these numbers is divisible by 3? (a) 2409 (b) 2413 (c) 2407 (d) 2411
18. 18.What is the units digit of 9^20? (a) 2 (b) 1 (c) 9 (d) 0
19. 19.What is the units digit of 4^57? (a) 4 (b) 5 (c) 3 (d) 6
20. 20.What is the units digit of 7^20? (a) 3 (b) 9 (c) 7 (d) 1
21. 21.What is the remainder when 54 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
22. 22.What is the units digit of 8^12? (a) 6 (b) 2 (c) 8 (d) 4
23. 23.What is the smallest number you must add to 247 to make it divisible by 6? (a) 3 (b) 1 (c) 5 (d) 4
24. 24.Which of these numbers is divisible by 8? (a) 3388 (b) 3399 (c) 3392 (d) 3403
25. 25.Which of these numbers is divisible by 11? (a) 1306 (b) 1290 (c) 1311 (d) 1298
26. 26.What is the smallest number you must add to 274 to make it divisible by 9? (a) 7 (b) 4 (c) 5 (d) 6
27. 27.What is the smallest number you must add to 234 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
28. 28.What is the remainder when 115 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
29. 29.Which of these numbers is divisible by 11? (a) 7164 (b) 7154 (c) 7161 (d) 7177
30. 30.What is the units digit of 7^10? (a) 1 (b) 3 (c) 9 (d) 7
31. 31.What is the units digit of 9^24? (a) 9 (b) 0 (c) 1 (d) 2
32. 32.What is the smallest number you must add to 695 to make it divisible by 12? (a) 2 (b) 11 (c) 1 (d) 3
33. 33.Which of these numbers is divisible by 11? (a) 4449 (b) 4465 (c) 4455 (d) 4468
34. 34.What is the remainder when 474 is divided by 11? (a) 10 (b) 2 (c) 0 (d) 1
35. 35.What is the smallest number you must add to 44 to make it divisible by 3? (a) 1 (b) 2
36. 36.What is the remainder when 442 is divided by 9? (a) 0 (b) 8 (c) 1 (d) 2
37. 37.What is the remainder when 396 is divided by 5? (a) 2 (b) 0 (c) 1 (d) 4
38. 38.What is the remainder when 97 is divided by 3? (a) 1 (b) 2 (c) 0
39. 39.What is the remainder when 565 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 2
40. 40.What is the smallest number you must add to 103 to make it divisible by 3? (a) 2 (b) 1

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