Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 813 · 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 220 is divided by 3? (a) 1 (b) 0 (c) 2
2. 2.What is the remainder when 288 is divided by 10? (a) 7 (b) 2 (c) 8 (d) 9
3. 3.What is the units digit of 2^54? (a) 4 (b) 8 (c) 6 (d) 2
4. 4.Which of these numbers is divisible by 6? (a) 2451 (b) 2448 (c) 2444 (d) 2459
5. 5.What is the remainder when 187 is divided by 3? (a) 2 (b) 1 (c) 0
6. 6.What is the units digit of 8^13? (a) 6 (b) 4 (c) 8 (d) 2
7. 7.What is the remainder when 446 is divided by 8? (a) 6 (b) 5 (c) 2 (d) 7
8. 8.Which of these numbers is divisible by 3? (a) 1291 (b) 1286 (c) 1288 (d) 1287
9. 9.What is the remainder when 145 is divided by 8? (a) 0 (b) 2 (c) 1 (d) 7
10. 10.Which of these numbers is divisible by 6? (a) 2038 (b) 2034 (c) 2041 (d) 2033
11. 11.What is the smallest number you must add to 505 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 1
12. 12.What is the remainder when 279 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
13. 13.What is the remainder when 309 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
14. 14.What is the remainder when 172 is divided by 12? (a) 4 (b) 8 (c) 3 (d) 5
15. 15.What is the units digit of 8^15? (a) 4 (b) 6 (c) 8 (d) 2
16. 16.What is the smallest number you must add to 655 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 7
17. 17.What is the smallest number you must add to 459 to make it divisible by 8? (a) 3 (b) 5 (c) 6 (d) 4
18. 18.What is the remainder when 394 is divided by 12? (a) 11 (b) 10 (c) 2 (d) 9
19. 19.What is the smallest number you must add to 692 to make it divisible by 10? (a) 8 (b) 9 (c) 2 (d) 7
20. 20.What is the remainder when 199 is divided by 3? (a) 2 (b) 1 (c) 0
21. 21.What is the remainder when 422 is divided by 12? (a) 10 (b) 2 (c) 1 (d) 3
22. 22.Which of these numbers is divisible by 11? (a) 660 (b) 673 (c) 653 (d) 666
23. 23.What is the smallest number you must add to 197 to make it divisible by 12? (a) 6 (b) 8 (c) 7 (d) 5
24. 24.What is the units digit of 2^11? (a) 6 (b) 8 (c) 2 (d) 4
25. 25.Which of these numbers is divisible by 4? (a) 3003 (b) 3004 (c) 3005 (d) 3009
26. 26.What is the smallest number you must add to 257 to make it divisible by 7? (a) 5 (b) 3 (c) 1 (d) 2
27. 27.Which of these numbers is divisible by 9? (a) 1464 (b) 1458 (c) 1475 (d) 1456
28. 28.What is the remainder when 171 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
29. 29.What is the remainder when 157 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
30. 30.Which of these numbers is divisible by 4? (a) 3312 (b) 3313 (c) 3319 (d) 3310
31. 31.Which of these numbers is divisible by 4? (a) 3400 (b) 3406 (c) 3401 (d) 3399
32. 32.What is the units digit of 2^51? (a) 8 (b) 2 (c) 4 (d) 6
33. 33.What is the units digit of 3^23? (a) 3 (b) 9 (c) 1 (d) 7
34. 34.What is the units digit of 9^23? (a) 1 (b) 9 (c) 8 (d) 7
35. 35.Which of these numbers is divisible by 11? (a) 1490 (b) 1474 (c) 1472 (d) 1481
36. 36.What is the smallest number you must add to 779 to make it divisible by 12? (a) 3 (b) 11 (c) 2 (d) 1
37. 37.What is the units digit of 9^38? (a) 0 (b) 1 (c) 2 (d) 9
38. 38.What is the remainder when 331 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
39. 39.What is the smallest number you must add to 306 to make it divisible by 4? (a) 1 (b) 3 (c) 2
40. 40.What is the units digit of 8^41? (a) 6 (b) 2 (c) 8 (d) 4

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