Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 490 · 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 remainder when 515 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 3
2. 2.Which of these numbers is divisible by 4? (a) 1084 (b) 1083 (c) 1086 (d) 1090
3. 3.What is the remainder when 494 is divided by 12? (a) 2 (b) 3 (c) 1 (d) 10
4. 4.Which of these numbers is divisible by 8? (a) 6455 (b) 6445 (c) 6448 (d) 6458
5. 5.What is the units digit of 8^24? (a) 4 (b) 8 (c) 6 (d) 2
6. 6.What is the smallest number you must add to 332 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
7. 7.What is the units digit of 3^60? (a) 9 (b) 1 (c) 3 (d) 7
8. 8.Which of these numbers is divisible by 6? (a) 2545 (b) 2537 (c) 2542 (d) 2538
9. 9.What is the smallest number you must add to 374 to make it divisible by 8? (a) 6 (b) 2 (c) 1 (d) 3
10. 10.Which of these numbers is divisible by 6? (a) 4214 (b) 4221 (c) 4218 (d) 4225
11. 11.What is the remainder when 213 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
12. 12.What is the units digit of 9^52? (a) 9 (b) 2 (c) 0 (d) 1
13. 13.What is the remainder when 425 is divided by 7? (a) 6 (b) 4 (c) 2 (d) 5
14. 14.Which of these numbers is divisible by 6? (a) 2638 (b) 2650 (c) 2640 (d) 2645
15. 15.Which of these numbers is divisible by 8? (a) 1646 (b) 1649 (c) 1648 (d) 1659
16. 16.Which of these numbers is divisible by 6? (a) 5005 (b) 5013 (c) 5003 (d) 5004
17. 17.What is the units digit of 8^21? (a) 8 (b) 6 (c) 2 (d) 4
18. 18.What is the smallest number you must add to 693 to make it divisible by 10? (a) 7 (b) 8 (c) 3 (d) 6
19. 19.What is the remainder when 423 is divided by 10? (a) 4 (b) 7 (c) 2 (d) 3
20. 20.What is the remainder when 87 is divided by 7? (a) 4 (b) 5 (c) 2 (d) 3
21. 21.What is the units digit of 8^47? (a) 8 (b) 6 (c) 2 (d) 4
22. 22.What is the units digit of 2^24? (a) 4 (b) 6 (c) 2 (d) 8
23. 23.What is the remainder when 644 is divided by 8? (a) 5 (b) 6 (c) 3 (d) 4
24. 24.What is the remainder when 926 is divided by 12? (a) 2 (b) 1 (c) 3 (d) 10
25. 25.What is the smallest number you must add to 177 to make it divisible by 4? (a) 1 (b) 2 (c) 3
26. 26.What is the smallest number you must add to 208 to make it divisible by 10? (a) 3 (b) 8 (c) 2 (d) 1
27. 27.Which of these numbers is divisible by 3? (a) 1944 (b) 1942 (c) 1948 (d) 1946
28. 28.What is the smallest number you must add to 254 to make it divisible by 11? (a) 10 (b) 1 (c) 9 (d) 8
29. 29.What is the units digit of 7^22? (a) 7 (b) 9 (c) 3 (d) 1
30. 30.What is the smallest number you must add to 237 to make it divisible by 6? (a) 5 (b) 4 (c) 2 (d) 3
31. 31.What is the units digit of 2^30? (a) 4 (b) 6 (c) 8 (d) 2
32. 32.What is the smallest number you must add to 939 to make it divisible by 12? (a) 10 (b) 8 (c) 9 (d) 3
33. 33.What is the units digit of 3^30? (a) 9 (b) 1 (c) 3 (d) 7
34. 34.What is the remainder when 164 is divided by 3? (a) 1 (b) 2 (c) 0
35. 35.Which of these numbers is divisible by 6? (a) 1700 (b) 1694 (c) 1698 (d) 1708
36. 36.What is the smallest number you must add to 148 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
37. 37.What is the smallest number you must add to 136 to make it divisible by 3? (a) 1 (b) 2
38. 38.What is the remainder when 302 is divided by 12? (a) 2 (b) 3 (c) 1 (d) 10
39. 39.What is the units digit of 4^46? (a) 5 (b) 4 (c) 7 (d) 6
40. 40.What is the remainder when 210 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2

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