Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 771 · 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 241 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
2. 2.Which of these numbers is divisible by 8? (a) 3040 (b) 3052 (c) 3045 (d) 3038
3. 3.Which of these numbers is divisible by 6? (a) 1705 (b) 1714 (c) 1699 (d) 1704
4. 4.What is the units digit of 2^23? (a) 2 (b) 4 (c) 8 (d) 6
5. 5.What is the units digit of 9^49? (a) 9 (b) 1 (c) 8 (d) 7
6. 6.What is the smallest number you must add to 495 to make it divisible by 7? (a) 3 (b) 5 (c) 1 (d) 2
7. 7.What is the remainder when 394 is divided by 8? (a) 2 (b) 1 (c) 3 (d) 6
8. 8.What is the smallest number you must add to 513 to make it divisible by 8? (a) 5 (b) 1 (c) 6 (d) 7
9. 9.Which of these numbers is divisible by 11? (a) 984 (b) 1004 (c) 992 (d) 990
10. 10.What is the remainder when 369 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
11. 11.Which of these numbers is divisible by 4? (a) 843 (b) 849 (c) 846 (d) 844
12. 12.What is the remainder when 574 is divided by 10? (a) 3 (b) 5 (c) 4 (d) 6
13. 13.What is the units digit of 2^10? (a) 2 (b) 8 (c) 4 (d) 6
14. 14.What is the smallest number you must add to 164 to make it divisible by 11? (a) 3 (b) 1 (c) 10 (d) 2
15. 15.Which of these numbers is divisible by 6? (a) 4206 (b) 4217 (c) 4211 (d) 4205
16. 16.What is the remainder when 503 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 1
17. 17.What is the smallest number you must add to 223 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
18. 18.Which of these numbers is divisible by 11? (a) 1299 (b) 1297 (c) 1277 (d) 1287
19. 19.What is the smallest number you must add to 279 to make it divisible by 10? (a) 2 (b) 3 (c) 9 (d) 1
20. 20.What is the remainder when 238 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
21. 21.Which of these numbers is divisible by 9? (a) 2518 (b) 2520 (c) 2533 (d) 2522
22. 22.What is the smallest number you must add to 698 to make it divisible by 11? (a) 5 (b) 7 (c) 8 (d) 6
23. 23.Which of these numbers is divisible by 6? (a) 3562 (b) 3564 (c) 3571 (d) 3566
24. 24.What is the remainder when 655 is divided by 9? (a) 2 (b) 8 (c) 6 (d) 7
25. 25.Which of these numbers is divisible by 6? (a) 4139 (b) 4128 (c) 4130 (d) 4124
26. 26.What is the units digit of 4^46? (a) 4 (b) 5 (c) 6 (d) 7
27. 27.What is the units digit of 3^20? (a) 1 (b) 3 (c) 7 (d) 9
28. 28.What is the remainder when 64 is divided by 3? (a) 2 (b) 1 (c) 0
29. 29.What is the units digit of 9^18? (a) 1 (b) 9 (c) 2 (d) 0
30. 30.Which of these numbers is divisible by 3? (a) 392 (b) 385 (c) 387 (d) 388
31. 31.Which of these numbers is divisible by 8? (a) 3804 (b) 3818 (c) 3808 (d) 3811
32. 32.What is the units digit of 7^17? (a) 3 (b) 1 (c) 7 (d) 9
33. 33.What is the remainder when 149 is divided by 7? (a) 5 (b) 1 (c) 3 (d) 2
34. 34.Which of these numbers is divisible by 6? (a) 1304 (b) 1308 (c) 1311 (d) 1318
35. 35.Which of these numbers is divisible by 6? (a) 3751 (b) 3749 (c) 3739 (d) 3744
36. 36.What is the smallest number you must add to 399 to make it divisible by 12? (a) 3 (b) 9 (c) 10 (d) 8
37. 37.Which of these numbers is divisible by 11? (a) 7563 (b) 7572 (c) 7552 (d) 7557
38. 38.What is the remainder when 696 is divided by 11? (a) 3 (b) 2 (c) 4 (d) 8
39. 39.What is the units digit of 2^55? (a) 4 (b) 2 (c) 6 (d) 8
40. 40.What is the smallest number you must add to 871 to make it divisible by 12? (a) 6 (b) 7 (c) 5 (d) 4

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