Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 706 · 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 units digit of 2^57? (a) 8 (b) 4 (c) 6 (d) 2
2. 2.What is the units digit of 2^47? (a) 6 (b) 2 (c) 8 (d) 4
3. 3.What is the units digit of 9^10? (a) 2 (b) 9 (c) 1 (d) 0
4. 4.What is the smallest number you must add to 423 to make it divisible by 12? (a) 3 (b) 10 (c) 9 (d) 8
5. 5.What is the smallest number you must add to 145 to make it divisible by 3? (a) 2 (b) 1
6. 6.What is the smallest number you must add to 838 to make it divisible by 11? (a) 10 (b) 8 (c) 2 (d) 9
7. 7.What is the smallest number you must add to 43 to make it divisible by 3? (a) 1 (b) 2
8. 8.What is the remainder when 321 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
9. 9.Which of these numbers is divisible by 11? (a) 8921 (b) 8924 (c) 8919 (d) 8942
10. 10.Which of these numbers is divisible by 3? (a) 1381 (b) 1387 (c) 1384 (d) 1383
11. 11.What is the smallest number you must add to 743 to make it divisible by 10? (a) 8 (b) 3 (c) 7 (d) 6
12. 12.Which of these numbers is divisible by 6? (a) 4782 (b) 4777 (c) 4790 (d) 4784
13. 13.What is the units digit of 9^21? (a) 1 (b) 7 (c) 8 (d) 9
14. 14.What is the remainder when 350 is divided by 11? (a) 9 (b) 8 (c) 2 (d) 10
15. 15.Which of these numbers is divisible by 3? (a) 2660 (b) 2655 (c) 2654 (d) 2656
16. 16.What is the remainder when 144 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 3
17. 17.Which of these numbers is divisible by 3? (a) 2297 (b) 2293 (c) 2300 (d) 2295
18. 18.Which of these numbers is divisible by 4? (a) 629 (b) 627 (c) 633 (d) 628
19. 19.What is the smallest number you must add to 476 to make it divisible by 8? (a) 3 (b) 5 (c) 4 (d) 6
20. 20.Which of these numbers is divisible by 4? (a) 2234 (b) 2239 (c) 2231 (d) 2232
21. 21.Which of these numbers is divisible by 8? (a) 6057 (b) 6049 (c) 6047 (d) 6048
22. 22.What is the units digit of 2^13? (a) 6 (b) 2 (c) 4 (d) 8
23. 23.Which of these numbers is divisible by 8? (a) 5768 (b) 5779 (c) 5762 (d) 5771
24. 24.What is the remainder when 353 is divided by 10? (a) 3 (b) 4 (c) 2 (d) 7
25. 25.What is the remainder when 211 is divided by 12? (a) 6 (b) 5 (c) 7 (d) 8
26. 26.What is the units digit of 7^49? (a) 9 (b) 3 (c) 1 (d) 7
27. 27.Which of these numbers is divisible by 4? (a) 2409 (b) 2406 (c) 2401 (d) 2404
28. 28.What is the remainder when 743 is divided by 11? (a) 6 (b) 7 (c) 8 (d) 5
29. 29.What is the smallest number you must add to 90 to make it divisible by 7? (a) 3 (b) 1 (c) 6 (d) 2
30. 30.What is the remainder when 199 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
31. 31.What is the smallest number you must add to 429 to make it divisible by 9? (a) 4 (b) 3 (c) 6 (d) 2
32. 32.What is the remainder when 366 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
33. 33.What is the smallest number you must add to 234 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
34. 34.What is the remainder when 212 is divided by 10? (a) 8 (b) 2 (c) 1 (d) 3
35. 35.Which of these numbers is divisible by 3? (a) 2620 (b) 2623 (c) 2619 (d) 2618
36. 36.What is the smallest number you must add to 66 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
37. 37.What is the remainder when 294 is divided by 11? (a) 7 (b) 3 (c) 8 (d) 9
38. 38.What is the units digit of 4^55? (a) 5 (b) 3 (c) 6 (d) 4
39. 39.What is the smallest number you must add to 403 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
40. 40.What is the smallest number you must add to 181 to make it divisible by 4? (a) 3 (b) 1 (c) 2

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