Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 597 · 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 289 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
2. 2.What is the smallest number you must add to 100 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 2
3. 3.What is the smallest number you must add to 242 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the units digit of 7^40? (a) 9 (b) 7 (c) 3 (d) 1
5. 5.What is the remainder when 366 is divided by 7? (a) 3 (b) 2 (c) 5 (d) 1
6. 6.What is the units digit of 3^15? (a) 1 (b) 3 (c) 9 (d) 7
7. 7.Which of these numbers is divisible by 11? (a) 1398 (b) 1429 (c) 1408 (d) 1418
8. 8.Which of these numbers is divisible by 6? (a) 769 (b) 779 (c) 785 (d) 774
9. 9.What is the remainder when 78 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
10. 10.Which of these numbers is divisible by 4? (a) 337 (b) 332 (c) 330 (d) 335
11. 11.What is the units digit of 8^36? (a) 6 (b) 2 (c) 4 (d) 8
12. 12.Which of these numbers is divisible by 6? (a) 1344 (b) 1351 (c) 1341 (d) 1346
13. 13.What is the smallest number you must add to 567 to make it divisible by 11? (a) 5 (b) 6 (c) 4 (d) 7
14. 14.What is the units digit of 7^23? (a) 9 (b) 7 (c) 1 (d) 3
15. 15.What is the remainder when 292 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
16. 16.What is the remainder when 796 is divided by 12? (a) 5 (b) 4 (c) 3 (d) 8
17. 17.Which of these numbers is divisible by 8? (a) 5950 (b) 5944 (c) 5939 (d) 5954
18. 18.What is the smallest number you must add to 258 to make it divisible by 7? (a) 2 (b) 3 (c) 1 (d) 6
19. 19.What is the remainder when 197 is divided by 7? (a) 2 (b) 0 (c) 6 (d) 1
20. 20.What is the units digit of 9^53? (a) 9 (b) 8 (c) 7 (d) 1
21. 21.What is the remainder when 58 is divided by 3? (a) 2 (b) 0 (c) 1
22. 22.What is the units digit of 7^36? (a) 3 (b) 7 (c) 1 (d) 9
23. 23.What is the units digit of 4^55? (a) 4 (b) 6 (c) 5 (d) 3
24. 24.What is the smallest number you must add to 454 to make it divisible by 6? (a) 2 (b) 4 (c) 1 (d) 3
25. 25.What is the units digit of 2^24? (a) 6 (b) 2 (c) 8 (d) 4
26. 26.What is the units digit of 4^30? (a) 4 (b) 7 (c) 5 (d) 6
27. 27.What is the smallest number you must add to 691 to make it divisible by 11? (a) 9 (b) 2 (c) 1 (d) 3
28. 28.What is the remainder when 346 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 0
29. 29.What is the units digit of 3^39? (a) 9 (b) 7 (c) 1 (d) 3
30. 30.Which of these numbers is divisible by 4? (a) 2675 (b) 2669 (c) 2668 (d) 2666
31. 31.What is the smallest number you must add to 88 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
32. 32.What is the units digit of 8^59? (a) 4 (b) 2 (c) 8 (d) 6
33. 33.What is the smallest number you must add to 153 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
34. 34.Which of these numbers is divisible by 8? (a) 4624 (b) 4633 (c) 4625 (d) 4620
35. 35.Which of these numbers is divisible by 6? (a) 4992 (b) 4997 (c) 5003 (d) 4989
36. 36.What is the smallest number you must add to 498 to make it divisible by 8? (a) 2 (b) 7 (c) 5 (d) 6
37. 37.What is the smallest number you must add to 604 to make it divisible by 12? (a) 4 (b) 9 (c) 8 (d) 7
38. 38.What is the remainder when 325 is divided by 12? (a) 1 (b) 2 (c) 0 (d) 11
39. 39.What is the units digit of 7^18? (a) 3 (b) 7 (c) 1 (d) 9
40. 40.What is the smallest number you must add to 662 to make it divisible by 9? (a) 5 (b) 6 (c) 3 (d) 4

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