Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 85 · 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 8^25? (a) 8 (b) 4 (c) 2 (d) 6
2. 2.What is the smallest number you must add to 133 to make it divisible by 10? (a) 6 (b) 3 (c) 7 (d) 8
3. 3.What is the smallest number you must add to 111 to make it divisible by 9? (a) 7 (b) 5 (c) 6 (d) 3
4. 4.What is the smallest number you must add to 297 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 3
5. 5.What is the smallest number you must add to 213 to make it divisible by 12? (a) 9 (b) 2 (c) 3 (d) 4
6. 6.What is the remainder when 92 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
7. 7.Which of these numbers is divisible by 11? (a) 5540 (b) 5549 (c) 5544 (d) 5561
8. 8.Which of these numbers is divisible by 11? (a) 2244 (b) 2243 (c) 2256 (d) 2246
9. 9.What is the smallest number you must add to 165 to make it divisible by 4? (a) 3 (b) 1 (c) 2
10. 10.What is the units digit of 9^26? (a) 0 (b) 2 (c) 9 (d) 1
11. 11.What is the remainder when 127 is divided by 7? (a) 1 (b) 2 (c) 6 (d) 0
12. 12.What is the units digit of 7^44? (a) 3 (b) 9 (c) 1 (d) 7
13. 13.What is the remainder when 217 is divided by 8? (a) 2 (b) 7 (c) 1 (d) 0
14. 14.What is the units digit of 4^40? (a) 5 (b) 6 (c) 7 (d) 4
15. 15.What is the remainder when 435 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
16. 16.What is the smallest number you must add to 63 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 2
17. 17.Which of these numbers is divisible by 3? (a) 1882 (b) 1880 (c) 1881 (d) 1885
18. 18.Which of these numbers is divisible by 8? (a) 5180 (b) 5197 (c) 5184 (d) 5190
19. 19.What is the units digit of 3^18? (a) 3 (b) 7 (c) 9 (d) 1
20. 20.What is the smallest number you must add to 395 to make it divisible by 9? (a) 8 (b) 3 (c) 2 (d) 1
21. 21.Which of these numbers is divisible by 3? (a) 134 (b) 131 (c) 127 (d) 129
22. 22.What is the remainder when 336 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
23. 23.Which of these numbers is divisible by 6? (a) 463 (b) 472 (c) 468 (d) 475
24. 24.Which of these numbers is divisible by 6? (a) 3648 (b) 3656 (c) 3645 (d) 3653
25. 25.What is the remainder when 156 is divided by 11? (a) 2 (b) 3 (c) 1 (d) 9
26. 26.What is the units digit of 3^39? (a) 1 (b) 9 (c) 3 (d) 7
27. 27.Which of these numbers is divisible by 8? (a) 6415 (b) 6416 (c) 6418 (d) 6429
28. 28.What is the smallest number you must add to 75 to make it divisible by 4? (a) 3 (b) 2 (c) 1
29. 29.Which of these numbers is divisible by 4? (a) 244 (b) 241 (c) 247 (d) 249
30. 30.What is the smallest number you must add to 142 to make it divisible by 3? (a) 1 (b) 2
31. 31.What is the units digit of 9^46? (a) 0 (b) 2 (c) 1 (d) 9
32. 32.Which of these numbers is divisible by 3? (a) 1740 (b) 1741 (c) 1745 (d) 1739
33. 33.What is the units digit of 2^33? (a) 8 (b) 6 (c) 2 (d) 4
34. 34.Which of these numbers is divisible by 6? (a) 1362 (b) 1365 (c) 1357 (d) 1373
35. 35.What is the smallest number you must add to 226 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
36. 36.What is the smallest number you must add to 400 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
37. 37.What is the smallest number you must add to 64 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
38. 38.What is the remainder when 68 is divided by 3? (a) 1 (b) 0 (c) 2
39. 39.Which of these numbers is divisible by 11? (a) 3516 (b) 3499 (c) 3490 (d) 3498
40. 40.What is the units digit of 8^24? (a) 2 (b) 4 (c) 8 (d) 6

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