Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 680 · 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 units digit of 4^12? (a) 5 (b) 6 (c) 4 (d) 7
2. 2.What is the smallest number you must add to 149 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 2
3. 3.Which of these numbers is divisible by 11? (a) 2280 (b) 2303 (c) 2296 (d) 2288
4. 4.Which of these numbers is divisible by 6? (a) 3337 (b) 3332 (c) 3336 (d) 3347
5. 5.What is the units digit of 3^36? (a) 1 (b) 9 (c) 7 (d) 3
6. 6.What is the smallest number you must add to 156 to make it divisible by 11? (a) 2 (b) 9 (c) 8 (d) 10
7. 7.What is the units digit of 8^56? (a) 2 (b) 4 (c) 8 (d) 6
8. 8.What is the units digit of 2^25? (a) 6 (b) 8 (c) 4 (d) 2
9. 9.What is the remainder when 669 is divided by 12? (a) 10 (b) 8 (c) 3 (d) 9
10. 10.What is the remainder when 762 is divided by 10? (a) 2 (b) 1 (c) 3 (d) 8
11. 11.What is the remainder when 476 is divided by 6? (a) 2 (b) 3 (c) 1 (d) 4
12. 12.What is the smallest number you must add to 125 to make it divisible by 7? (a) 2 (b) 1 (c) 6 (d) 3
13. 13.Which of these numbers is divisible by 4? (a) 741 (b) 739 (c) 740 (d) 747
14. 14.What is the remainder when 49 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
15. 15.What is the units digit of 2^31? (a) 4 (b) 2 (c) 8 (d) 6
16. 16.What is the units digit of 3^53? (a) 3 (b) 9 (c) 1 (d) 7
17. 17.Which of these numbers is divisible by 3? (a) 1147 (b) 1144 (c) 1146 (d) 1150
18. 18.What is the units digit of 7^26? (a) 7 (b) 1 (c) 3 (d) 9
19. 19.Which of these numbers is divisible by 4? (a) 2554 (b) 2548 (c) 2545 (d) 2549
20. 20.What is the remainder when 224 is divided by 6? (a) 2 (b) 3 (c) 1 (d) 4
21. 21.Which of these numbers is divisible by 9? (a) 6310 (b) 6303 (c) 6309 (d) 6325
22. 22.Which of these numbers is divisible by 4? (a) 2543 (b) 2551 (c) 2546 (d) 2544
23. 23.What is the units digit of 9^54? (a) 2 (b) 1 (c) 9 (d) 0
24. 24.Which of these numbers is divisible by 4? (a) 3428 (b) 3431 (c) 3427 (d) 3433
25. 25.What is the remainder when 310 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
26. 26.What is the remainder when 115 is divided by 3? (a) 2 (b) 1 (c) 0
27. 27.What is the smallest number you must add to 177 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
28. 28.What is the remainder when 222 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 6
29. 29.Which of these numbers is divisible by 11? (a) 4521 (b) 4518 (c) 4540 (d) 4525
30. 30.Which of these numbers is divisible by 9? (a) 4166 (b) 4149 (c) 4156 (d) 4146
31. 31.What is the units digit of 9^12? (a) 9 (b) 1 (c) 0 (d) 2
32. 32.What is the units digit of 8^13? (a) 4 (b) 6 (c) 2 (d) 8
33. 33.Which of these numbers is divisible by 8? (a) 6747 (b) 6732 (c) 6736 (d) 6743
34. 34.What is the units digit of 4^30? (a) 6 (b) 5 (c) 7 (d) 4
35. 35.What is the smallest number you must add to 547 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
36. 36.What is the smallest number you must add to 164 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
37. 37.What is the remainder when 244 is divided by 10? (a) 3 (b) 6 (c) 5 (d) 4
38. 38.What is the smallest number you must add to 455 to make it divisible by 12? (a) 2 (b) 1 (c) 3 (d) 11
39. 39.What is the smallest number you must add to 46 to make it divisible by 3? (a) 1 (b) 2
40. 40.Which of these numbers is divisible by 4? (a) 1031 (b) 1023 (c) 1024 (d) 1026

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