Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 482 · 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 278 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
2. 2.What is the units digit of 8^25? (a) 4 (b) 8 (c) 6 (d) 2
3. 3.What is the remainder when 477 is divided by 6? (a) 3 (b) 2 (c) 5 (d) 4
4. 4.What is the units digit of 9^46? (a) 2 (b) 1 (c) 9 (d) 0
5. 5.What is the remainder when 118 is divided by 3? (a) 1 (b) 0 (c) 2
6. 6.Which of these numbers is divisible by 8? (a) 2841 (b) 2833 (c) 2832 (d) 2829
7. 7.Which of these numbers is divisible by 4? (a) 1112 (b) 1115 (c) 1118 (d) 1111
8. 8.What is the remainder when 222 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 2
9. 9.What is the smallest number you must add to 328 to make it divisible by 6? (a) 1 (b) 2 (c) 3 (d) 4
10. 10.What is the smallest number you must add to 460 to make it divisible by 12? (a) 7 (b) 4 (c) 9 (d) 8
11. 11.What is the smallest number you must add to 269 to make it divisible by 4? (a) 1 (b) 2 (c) 3
12. 12.What is the units digit of 4^55? (a) 4 (b) 3 (c) 5 (d) 6
13. 13.What is the remainder when 200 is divided by 9? (a) 7 (b) 3 (c) 1 (d) 2
14. 14.What is the units digit of 7^13? (a) 9 (b) 1 (c) 7 (d) 3
15. 15.Which of these numbers is divisible by 3? (a) 2177 (b) 2174 (c) 2175 (d) 2179
16. 16.What is the remainder when 686 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
17. 17.What is the units digit of 7^42? (a) 7 (b) 3 (c) 9 (d) 1
18. 18.Which of these numbers is divisible by 8? (a) 3141 (b) 3130 (c) 3128 (d) 3127
19. 19.What is the smallest number you must add to 113 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 2
20. 20.What is the units digit of 9^45? (a) 8 (b) 7 (c) 9 (d) 1
21. 21.What is the units digit of 7^58? (a) 1 (b) 3 (c) 9 (d) 7
22. 22.What is the smallest number you must add to 302 to make it divisible by 4? (a) 1 (b) 2 (c) 3
23. 23.Which of these numbers is divisible by 8? (a) 5814 (b) 5816 (c) 5823 (d) 5831
24. 24.What is the units digit of 8^31? (a) 8 (b) 4 (c) 2 (d) 6
25. 25.Which of these numbers is divisible by 4? (a) 1714 (b) 1705 (c) 1708 (d) 1709
26. 26.Which of these numbers is divisible by 3? (a) 1531 (b) 1526 (c) 1527 (d) 1529
27. 27.Which of these numbers is divisible by 6? (a) 3726 (b) 3727 (c) 3733 (d) 3722
28. 28.What is the smallest number you must add to 290 to make it divisible by 11? (a) 6 (b) 7 (c) 8 (d) 4
29. 29.What is the smallest number you must add to 106 to make it divisible by 7? (a) 1 (b) 6 (c) 4 (d) 5
30. 30.What is the smallest number you must add to 222 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
31. 31.What is the remainder when 529 is divided by 8? (a) 2 (b) 7 (c) 1 (d) 0
32. 32.What is the smallest number you must add to 739 to make it divisible by 12? (a) 7 (b) 4 (c) 5 (d) 6
33. 33.What is the smallest number you must add to 676 to make it divisible by 10? (a) 3 (b) 4 (c) 5 (d) 6
34. 34.Which of these numbers is divisible by 3? (a) 1200 (b) 1205 (c) 1198 (d) 1202
35. 35.Which of these numbers is divisible by 9? (a) 5569 (b) 5560 (c) 5545 (d) 5553
36. 36.What is the units digit of 9^16? (a) 1 (b) 9 (c) 2 (d) 0
37. 37.What is the units digit of 4^23? (a) 3 (b) 4 (c) 6 (d) 5
38. 38.What is the smallest number you must add to 127 to make it divisible by 4? (a) 1 (b) 3 (c) 2
39. 39.What is the smallest number you must add to 403 to make it divisible by 7? (a) 5 (b) 4 (c) 3 (d) 2
40. 40.What is the remainder when 586 is divided by 11? (a) 3 (b) 4 (c) 8 (d) 2

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