Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 644 · 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 smallest number you must add to 79 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 1
2. 2.Which of these numbers is divisible by 11? (a) 5577 (b) 5595 (c) 5581 (d) 5570
3. 3.What is the remainder when 562 is divided by 8? (a) 6 (b) 1 (c) 3 (d) 2
4. 4.What is the remainder when 128 is divided by 12? (a) 4 (b) 9 (c) 7 (d) 8
5. 5.What is the smallest number you must add to 52 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
6. 6.Which of these numbers is divisible by 6? (a) 557 (b) 551 (c) 544 (d) 546
7. 7.What is the smallest number you must add to 127 to make it divisible by 8? (a) 2 (b) 1 (c) 7 (d) 3
8. 8.What is the smallest number you must add to 429 to make it divisible by 9? (a) 3 (b) 2 (c) 6 (d) 4
9. 9.What is the remainder when 322 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
10. 10.What is the remainder when 277 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
11. 11.Which of these numbers is divisible by 6? (a) 3930 (b) 3937 (c) 3927 (d) 3933
12. 12.Which of these numbers is divisible by 8? (a) 1170 (b) 1182 (c) 1168 (d) 1163
13. 13.What is the smallest number you must add to 524 to make it divisible by 7? (a) 6 (b) 1 (c) 3 (d) 2
14. 14.Which of these numbers is divisible by 4? (a) 2553 (b) 2545 (c) 2551 (d) 2548
15. 15.What is the units digit of 9^33? (a) 7 (b) 9 (c) 8 (d) 1
16. 16.Which of these numbers is divisible by 8? (a) 2346 (b) 2337 (c) 2344 (d) 2356
17. 17.What is the smallest number you must add to 77 to make it divisible by 3? (a) 1 (b) 2
18. 18.What is the units digit of 8^32? (a) 6 (b) 2 (c) 4 (d) 8
19. 19.Which of these numbers is divisible by 3? (a) 407 (b) 403 (c) 400 (d) 402
20. 20.Which of these numbers is divisible by 3? (a) 1015 (b) 1010 (c) 1011 (d) 1012
21. 21.What is the remainder when 212 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
22. 22.Which of these numbers is divisible by 3? (a) 1374 (b) 1375 (c) 1379 (d) 1372
23. 23.What is the units digit of 9^53? (a) 8 (b) 7 (c) 9 (d) 1
24. 24.What is the smallest number you must add to 154 to make it divisible by 3? (a) 2 (b) 1
25. 25.What is the units digit of 7^56? (a) 9 (b) 7 (c) 1 (d) 3
26. 26.Which of these numbers is divisible by 6? (a) 3222 (b) 3231 (c) 3226 (d) 3220
27. 27.What is the remainder when 109 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
28. 28.Which of these numbers is divisible by 4? (a) 539 (b) 542 (c) 540 (d) 545
29. 29.What is the smallest number you must add to 106 to make it divisible by 7? (a) 6 (b) 1 (c) 5 (d) 4
30. 30.What is the units digit of 9^10? (a) 2 (b) 0 (c) 1 (d) 9
31. 31.Which of these numbers is divisible by 4? (a) 2521 (b) 2525 (c) 2518 (d) 2520
32. 32.What is the smallest number you must add to 224 to make it divisible by 3? (a) 1 (b) 2
33. 33.What is the smallest number you must add to 430 to make it divisible by 7? (a) 6 (b) 3 (c) 4 (d) 5
34. 34.What is the remainder when 206 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4
35. 35.What is the smallest number you must add to 741 to make it divisible by 10? (a) 8 (b) 7 (c) 9 (d) 1
36. 36.What is the remainder when 301 is divided by 6? (a) 0 (b) 1 (c) 2 (d) 5
37. 37.Which of these numbers is divisible by 4? (a) 1259 (b) 1253 (c) 1252 (d) 1251
38. 38.What is the units digit of 7^21? (a) 1 (b) 7 (c) 3 (d) 9
39. 39.Which of these numbers is divisible by 6? (a) 369 (b) 382 (c) 372 (d) 375
40. 40.What is the remainder when 555 is divided by 11? (a) 6 (b) 7 (c) 5 (d) 4

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