Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 168 · 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 154 is divided by 8? (a) 1 (b) 6 (c) 2 (d) 3
2. 2.What is the units digit of 8^58? (a) 8 (b) 6 (c) 2 (d) 4
3. 3.Which of these numbers is divisible by 9? (a) 7112 (b) 7119 (c) 7121 (d) 7133
4. 4.What is the remainder when 416 is divided by 9? (a) 1 (b) 7 (c) 3 (d) 2
5. 5.Which of these numbers is divisible by 6? (a) 5179 (b) 5178 (c) 5176 (d) 5185
6. 6.What is the units digit of 9^17? (a) 1 (b) 8 (c) 9 (d) 7
7. 7.What is the remainder when 208 is divided by 9? (a) 1 (b) 8 (c) 0 (d) 2
8. 8.Which of these numbers is divisible by 11? (a) 5256 (b) 5265 (c) 5242 (d) 5247
9. 9.What is the units digit of 7^12? (a) 3 (b) 7 (c) 9 (d) 1
10. 10.What is the units digit of 3^55? (a) 9 (b) 1 (c) 7 (d) 3
11. 11.What is the units digit of 7^50? (a) 3 (b) 1 (c) 9 (d) 7
12. 12.What is the units digit of 7^35? (a) 1 (b) 9 (c) 3 (d) 7
13. 13.Which of these numbers is divisible by 3? (a) 1050 (b) 1049 (c) 1051 (d) 1054
14. 14.Which of these numbers is divisible by 6? (a) 3655 (b) 3669 (c) 3660 (d) 3661
15. 15.Which of these numbers is divisible by 11? (a) 8189 (b) 8184 (c) 8205 (d) 8183
16. 16.What is the units digit of 9^42? (a) 1 (b) 0 (c) 9 (d) 2
17. 17.What is the remainder when 206 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
18. 18.What is the remainder when 213 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
19. 19.What is the smallest number you must add to 333 to make it divisible by 11? (a) 7 (b) 9 (c) 8 (d) 3
20. 20.What is the smallest number you must add to 783 to make it divisible by 10? (a) 8 (b) 7 (c) 6 (d) 3
21. 21.What is the units digit of 7^44? (a) 9 (b) 3 (c) 1 (d) 7
22. 22.What is the remainder when 128 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
23. 23.Which of these numbers is divisible by 8? (a) 6063 (b) 6064 (c) 6066 (d) 6078
24. 24.What is the smallest number you must add to 376 to make it divisible by 7? (a) 2 (b) 1 (c) 3 (d) 5
25. 25.What is the units digit of 7^55? (a) 3 (b) 9 (c) 7 (d) 1
26. 26.What is the remainder when 497 is divided by 8? (a) 7 (b) 1 (c) 2 (d) 0
27. 27.What is the smallest number you must add to 311 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
28. 28.Which of these numbers is divisible by 6? (a) 2941 (b) 2934 (c) 2933 (d) 2935
29. 29.Which of these numbers is divisible by 3? (a) 2117 (b) 2123 (c) 2119 (d) 2118
30. 30.Which of these numbers is divisible by 4? (a) 1496 (b) 1498 (c) 1495 (d) 1501
31. 31.What is the units digit of 9^30? (a) 1 (b) 0 (c) 9 (d) 2
32. 32.What is the units digit of 3^42? (a) 7 (b) 9 (c) 3 (d) 1
33. 33.What is the smallest number you must add to 184 to make it divisible by 12? (a) 4 (b) 7 (c) 9 (d) 8
34. 34.What is the remainder when 161 is divided by 3? (a) 0 (b) 1 (c) 2
35. 35.Which of these numbers is divisible by 9? (a) 5371 (b) 5378 (c) 5364 (d) 5362
36. 36.What is the units digit of 7^19? (a) 1 (b) 3 (c) 7 (d) 9
37. 37.Which of these numbers is divisible by 3? (a) 1584 (b) 1582 (c) 1586 (d) 1588
38. 38.What is the units digit of 2^25? (a) 4 (b) 8 (c) 2 (d) 6
39. 39.What is the smallest number you must add to 263 to make it divisible by 12? (a) 2 (b) 11 (c) 1 (d) 3
40. 40.Which of these numbers is divisible by 4? (a) 585 (b) 588 (c) 589 (d) 593

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