Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 776 · 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 177 is divided by 9? (a) 7 (b) 5 (c) 6 (d) 3
2. 2.What is the remainder when 212 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
3. 3.Which of these numbers is divisible by 4? (a) 916 (b) 921 (c) 917 (d) 914
4. 4.What is the smallest number you must add to 103 to make it divisible by 4? (a) 2 (b) 1 (c) 3
5. 5.What is the units digit of 9^45? (a) 7 (b) 1 (c) 8 (d) 9
6. 6.Which of these numbers is divisible by 11? (a) 2168 (b) 2155 (c) 2156 (d) 2162
7. 7.What is the remainder when 176 is divided by 3? (a) 1 (b) 2 (c) 0
8. 8.What is the remainder when 244 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
9. 9.What is the remainder when 312 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
10. 10.What is the smallest number you must add to 421 to make it divisible by 8? (a) 5 (b) 2 (c) 4 (d) 3
11. 11.What is the units digit of 3^37? (a) 3 (b) 7 (c) 1 (d) 9
12. 12.What is the remainder when 436 is divided by 10? (a) 5 (b) 7 (c) 4 (d) 6
13. 13.Which of these numbers is divisible by 8? (a) 3498 (b) 3504 (c) 3513 (d) 3505
14. 14.What is the units digit of 9^23? (a) 8 (b) 9 (c) 1 (d) 7
15. 15.What is the smallest number you must add to 183 to make it divisible by 9? (a) 7 (b) 5 (c) 3 (d) 6
16. 16.What is the smallest number you must add to 339 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
17. 17.What is the remainder when 144 is divided by 10? (a) 4 (b) 6 (c) 3 (d) 5
18. 18.What is the units digit of 2^41? (a) 8 (b) 4 (c) 6 (d) 2
19. 19.What is the remainder when 131 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
20. 20.Which of these numbers is divisible by 8? (a) 752 (b) 762 (c) 751 (d) 753
21. 21.What is the smallest number you must add to 218 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
22. 22.Which of these numbers is divisible by 3? (a) 1570 (b) 1569 (c) 1574 (d) 1568
23. 23.Which of these numbers is divisible by 3? (a) 1841 (b) 1836 (c) 1835 (d) 1837
24. 24.What is the smallest number you must add to 325 to make it divisible by 10? (a) 6 (b) 7 (c) 4 (d) 5
25. 25.Which of these numbers is divisible by 8? (a) 415 (b) 418 (c) 408 (d) 403
26. 26.What is the smallest number you must add to 357 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
27. 27.What is the remainder when 181 is divided by 10? (a) 0 (b) 9 (c) 1 (d) 2
28. 28.Which of these numbers is divisible by 6? (a) 3757 (b) 3750 (c) 3747 (d) 3751
29. 29.Which of these numbers is divisible by 8? (a) 2730 (b) 2741 (c) 2726 (d) 2728
30. 30.What is the units digit of 8^17? (a) 8 (b) 4 (c) 2 (d) 6
31. 31.What is the remainder when 267 is divided by 12? (a) 9 (b) 3 (c) 4 (d) 2
32. 32.What is the units digit of 3^31? (a) 3 (b) 1 (c) 9 (d) 7
33. 33.What is the remainder when 534 is divided by 9? (a) 3 (b) 4 (c) 6 (d) 2
34. 34.Which of these numbers is divisible by 4? (a) 3436 (b) 3435 (c) 3442 (d) 3439
35. 35.Which of these numbers is divisible by 4? (a) 2398 (b) 2401 (c) 2393 (d) 2396
36. 36.What is the smallest number you must add to 523 to make it divisible by 12? (a) 7 (b) 4 (c) 5 (d) 6
37. 37.What is the units digit of 3^42? (a) 3 (b) 1 (c) 9 (d) 7
38. 38.What is the units digit of 8^32? (a) 8 (b) 6 (c) 2 (d) 4
39. 39.What is the remainder when 691 is divided by 9? (a) 6 (b) 8 (c) 2 (d) 7
40. 40.Which of these numbers is divisible by 3? (a) 391 (b) 394 (c) 397 (d) 393

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