Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 427 · 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 491 is divided by 7? (a) 1 (b) 6 (c) 0 (d) 2
2. 2.What is the units digit of 8^36? (a) 8 (b) 6 (c) 2 (d) 4
3. 3.Which of these numbers is divisible by 9? (a) 6473 (b) 6493 (c) 6485 (d) 6480
4. 4.Which of these numbers is divisible by 11? (a) 6487 (b) 6493 (c) 6479 (d) 6478
5. 5.What is the smallest number you must add to 382 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
6. 6.What is the remainder when 158 is divided by 10? (a) 8 (b) 2 (c) 7 (d) 9
7. 7.Which of these numbers is divisible by 3? (a) 398 (b) 400 (c) 395 (d) 396
8. 8.Which of these numbers is divisible by 6? (a) 4442 (b) 4436 (c) 4440 (d) 4450
9. 9.What is the remainder when 634 is divided by 11? (a) 6 (b) 8 (c) 7 (d) 4
10. 10.What is the smallest number you must add to 167 to make it divisible by 3? (a) 1 (b) 2
11. 11.What is the remainder when 347 is divided by 10? (a) 3 (b) 7 (c) 8 (d) 6
12. 12.What is the smallest number you must add to 655 to make it divisible by 9? (a) 3 (b) 7 (c) 2 (d) 1
13. 13.Which of these numbers is divisible by 8? (a) 2026 (b) 2032 (c) 2044 (d) 2039
14. 14.What is the smallest number you must add to 327 to make it divisible by 12? (a) 8 (b) 9 (c) 3 (d) 10
15. 15.Which of these numbers is divisible by 4? (a) 933 (b) 938 (c) 936 (d) 942
16. 16.What is the remainder when 388 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
17. 17.What is the remainder when 208 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
18. 18.Which of these numbers is divisible by 3? (a) 127 (b) 133 (c) 129 (d) 130
19. 19.Which of these numbers is divisible by 11? (a) 8717 (b) 8709 (c) 8732 (d) 8712
20. 20.What is the units digit of 4^38? (a) 5 (b) 7 (c) 4 (d) 6
21. 21.What is the remainder when 242 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
22. 22.Which of these numbers is divisible by 3? (a) 1488 (b) 1489 (c) 1492 (d) 1487
23. 23.What is the units digit of 4^60? (a) 4 (b) 5 (c) 7 (d) 6
24. 24.What is the smallest number you must add to 127 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 1
25. 25.What is the smallest number you must add to 108 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
26. 26.What is the remainder when 130 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 3
27. 27.Which of these numbers is divisible by 4? (a) 1306 (b) 1308 (c) 1314 (d) 1311
28. 28.What is the remainder when 266 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 2
29. 29.Which of these numbers is divisible by 9? (a) 5026 (b) 5031 (c) 5045 (d) 5033
30. 30.What is the remainder when 395 is divided by 7? (a) 5 (b) 3 (c) 4 (d) 2
31. 31.What is the smallest number you must add to 548 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
32. 32.What is the smallest number you must add to 215 to make it divisible by 7? (a) 3 (b) 1 (c) 5 (d) 2
33. 33.What is the remainder when 339 is divided by 12? (a) 2 (b) 3 (c) 9 (d) 4
34. 34.Which of these numbers is divisible by 8? (a) 2051 (b) 2044 (c) 2058 (d) 2048
35. 35.What is the remainder when 211 is divided by 3? (a) 2 (b) 1 (c) 0
36. 36.Which of these numbers is divisible by 3? (a) 2273 (b) 2275 (c) 2269 (d) 2271
37. 37.Which of these numbers is divisible by 8? (a) 3415 (b) 3400 (c) 3393 (d) 3406
38. 38.What is the units digit of 4^43? (a) 3 (b) 6 (c) 4 (d) 5
39. 39.Which of these numbers is divisible by 8? (a) 1996 (b) 2004 (c) 2015 (d) 2000
40. 40.What is the units digit of 8^50? (a) 4 (b) 6 (c) 8 (d) 2

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