Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 214 · 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.Which of these numbers is divisible by 8? (a) 1197 (b) 1206 (c) 1192 (d) 1186
2. 2.What is the remainder when 547 is divided by 12? (a) 7 (b) 5 (c) 6 (d) 8
3. 3.What is the smallest number you must add to 350 to make it divisible by 12? (a) 9 (b) 10 (c) 11 (d) 2
4. 4.What is the units digit of 3^59? (a) 3 (b) 9 (c) 7 (d) 1
5. 5.What is the smallest number you must add to 317 to make it divisible by 4? (a) 2 (b) 3 (c) 1
6. 6.What is the smallest number you must add to 621 to make it divisible by 11? (a) 6 (b) 5 (c) 8 (d) 7
7. 7.What is the remainder when 273 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
8. 8.What is the remainder when 384 is divided by 11? (a) 1 (b) 8 (c) 10 (d) 9
9. 9.What is the remainder when 157 is divided by 9? (a) 3 (b) 5 (c) 4 (d) 6
10. 10.What is the units digit of 4^43? (a) 5 (b) 4 (c) 6 (d) 3
11. 11.What is the smallest number you must add to 523 to make it divisible by 12? (a) 6 (b) 4 (c) 7 (d) 5
12. 12.Which of these numbers is divisible by 3? (a) 670 (b) 664 (c) 668 (d) 666
13. 13.What is the units digit of 9^26? (a) 9 (b) 0 (c) 1 (d) 2
14. 14.What is the smallest number you must add to 369 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
15. 15.Which of these numbers is divisible by 3? (a) 868 (b) 870 (c) 874 (d) 871
16. 16.Which of these numbers is divisible by 6? (a) 411 (b) 415 (c) 407 (d) 408
17. 17.Which of these numbers is divisible by 11? (a) 2723 (b) 2734 (c) 2717 (d) 2710
18. 18.What is the remainder when 268 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
19. 19.What is the smallest number you must add to 367 to make it divisible by 8? (a) 7 (b) 1 (c) 3 (d) 2
20. 20.What is the remainder when 98 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
21. 21.What is the remainder when 238 is divided by 10? (a) 2 (b) 8 (c) 7 (d) 9
22. 22.Which of these numbers is divisible by 8? (a) 5275 (b) 5286 (c) 5280 (d) 5294
23. 23.What is the remainder when 194 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 2
24. 24.Which of these numbers is divisible by 3? (a) 909 (b) 908 (c) 911 (d) 914
25. 25.What is the remainder when 128 is divided by 9? (a) 2 (b) 3 (c) 1 (d) 7
26. 26.What is the remainder when 309 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
27. 27.What is the units digit of 3^15? (a) 7 (b) 3 (c) 9 (d) 1
28. 28.What is the smallest number you must add to 46 to make it divisible by 4? (a) 2 (b) 1 (c) 3
29. 29.What is the smallest number you must add to 437 to make it divisible by 8? (a) 4 (b) 3 (c) 2 (d) 5
30. 30.What is the remainder when 476 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 1
31. 31.What is the remainder when 75 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
32. 32.What is the remainder when 251 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
33. 33.What is the remainder when 532 is divided by 9? (a) 2 (b) 1 (c) 0 (d) 8
34. 34.What is the units digit of 7^52? (a) 1 (b) 7 (c) 3 (d) 9
35. 35.What is the remainder when 416 is divided by 12? (a) 4 (b) 7 (c) 9 (d) 8
36. 36.What is the remainder when 519 is divided by 11? (a) 1 (b) 9 (c) 3 (d) 2
37. 37.What is the smallest number you must add to 527 to make it divisible by 12? (a) 3 (b) 11 (c) 2 (d) 1
38. 38.What is the units digit of 9^11? (a) 7 (b) 9 (c) 8 (d) 1
39. 39.What is the remainder when 309 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
40. 40.What is the units digit of 7^38? (a) 1 (b) 9 (c) 7 (d) 3

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