Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 428 · 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 422 is divided by 10? (a) 2 (b) 3 (c) 1 (d) 8
2. 2.What is the units digit of 3^38? (a) 3 (b) 1 (c) 7 (d) 9
3. 3.Which of these numbers is divisible by 9? (a) 7706 (b) 7729 (c) 7713 (d) 7720
4. 4.Which of these numbers is divisible by 9? (a) 7343 (b) 7325 (c) 7328 (d) 7326
5. 5.What is the smallest number you must add to 235 to make it divisible by 3? (a) 1 (b) 2
6. 6.What is the units digit of 4^12? (a) 7 (b) 6 (c) 4 (d) 5
7. 7.What is the units digit of 8^33? (a) 2 (b) 4 (c) 8 (d) 6
8. 8.What is the units digit of 8^58? (a) 2 (b) 8 (c) 4 (d) 6
9. 9.What is the smallest number you must add to 145 to make it divisible by 12? (a) 1 (b) 11 (c) 9 (d) 10
10. 10.Which of these numbers is divisible by 4? (a) 1400 (b) 1397 (c) 1406 (d) 1402
11. 11.Which of these numbers is divisible by 6? (a) 2591 (b) 2592 (c) 2599 (d) 2593
12. 12.Which of these numbers is divisible by 11? (a) 3007 (b) 3020 (c) 3028 (d) 3014
13. 13.Which of these numbers is divisible by 6? (a) 2256 (b) 2253 (c) 2263 (d) 2260
14. 14.What is the smallest number you must add to 289 to make it divisible by 10? (a) 9 (b) 3 (c) 2 (d) 1
15. 15.What is the smallest number you must add to 541 to make it divisible by 9? (a) 7 (b) 6 (c) 1 (d) 8
16. 16.What is the smallest number you must add to 445 to make it divisible by 11? (a) 8 (b) 7 (c) 5 (d) 6
17. 17.What is the smallest number you must add to 319 to make it divisible by 4? (a) 3 (b) 2 (c) 1
18. 18.What is the units digit of 7^38? (a) 9 (b) 7 (c) 3 (d) 1
19. 19.What is the smallest number you must add to 88 to make it divisible by 3? (a) 1 (b) 2
20. 20.What is the units digit of 8^52? (a) 6 (b) 8 (c) 2 (d) 4
21. 21.What is the remainder when 688 is divided by 11? (a) 5 (b) 6 (c) 7 (d) 8
22. 22.What is the remainder when 327 is divided by 11? (a) 7 (b) 3 (c) 8 (d) 9
23. 23.What is the remainder when 441 is divided by 8? (a) 7 (b) 2 (c) 1 (d) 0
24. 24.What is the units digit of 8^41? (a) 6 (b) 2 (c) 8 (d) 4
25. 25.What is the remainder when 185 is divided by 3? (a) 1 (b) 0 (c) 2
26. 26.What is the remainder when 186 is divided by 12? (a) 8 (b) 7 (c) 6 (d) 5
27. 27.Which of these numbers is divisible by 11? (a) 2334 (b) 2330 (c) 2346 (d) 2332
28. 28.What is the remainder when 404 is divided by 11? (a) 7 (b) 8 (c) 3 (d) 9
29. 29.What is the smallest number you must add to 634 to make it divisible by 10? (a) 5 (b) 7 (c) 4 (d) 6
30. 30.Which of these numbers is divisible by 3? (a) 928 (b) 922 (c) 925 (d) 924
31. 31.What is the smallest number you must add to 174 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
32. 32.What is the remainder when 44 is divided by 3? (a) 0 (b) 1 (c) 2
33. 33.What is the remainder when 172 is divided by 11? (a) 4 (b) 6 (c) 8 (d) 7
34. 34.What is the units digit of 7^56? (a) 7 (b) 9 (c) 3 (d) 1
35. 35.Which of these numbers is divisible by 6? (a) 3612 (b) 3613 (c) 3611 (d) 3620
36. 36.What is the smallest number you must add to 83 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 2
37. 37.What is the units digit of 7^54? (a) 9 (b) 1 (c) 7 (d) 3
38. 38.What is the units digit of 2^17? (a) 4 (b) 6 (c) 2 (d) 8
39. 39.Which of these numbers is divisible by 3? (a) 1928 (b) 1925 (c) 1922 (d) 1923
40. 40.What is the units digit of 4^14? (a) 4 (b) 7 (c) 5 (d) 6

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