Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 414 · 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 9^40? (a) 9 (b) 2 (c) 1 (d) 0
2. 2.What is the remainder when 403 is divided by 10? (a) 4 (b) 2 (c) 3 (d) 7
3. 3.Which of these numbers is divisible by 8? (a) 6192 (b) 6191 (c) 6197 (d) 6201
4. 4.What is the smallest number you must add to 286 to make it divisible by 8? (a) 1 (b) 6 (c) 2 (d) 3
5. 5.What is the remainder when 153 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
6. 6.Which of these numbers is divisible by 4? (a) 2436 (b) 2437 (c) 2435 (d) 2441
7. 7.Which of these numbers is divisible by 3? (a) 644 (b) 647 (c) 645 (d) 650
8. 8.What is the units digit of 4^28? (a) 7 (b) 6 (c) 4 (d) 5
9. 9.What is the remainder when 61 is divided by 3? (a) 1 (b) 0 (c) 2
10. 10.What is the remainder when 590 is divided by 8? (a) 2 (b) 6 (c) 5 (d) 7
11. 11.What is the units digit of 8^35? (a) 6 (b) 2 (c) 8 (d) 4
12. 12.Which of these numbers is divisible by 6? (a) 4225 (b) 4216 (c) 4221 (d) 4218
13. 13.Which of these numbers is divisible by 11? (a) 8027 (b) 8049 (c) 8030 (d) 8039
14. 14.Which of these numbers is divisible by 8? (a) 3934 (b) 3936 (c) 3942 (d) 3946
15. 15.What is the remainder when 175 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 7
16. 16.Which of these numbers is divisible by 11? (a) 525 (b) 509 (c) 500 (d) 506
17. 17.What is the smallest number you must add to 223 to make it divisible by 8? (a) 3 (b) 7 (c) 2 (d) 1
18. 18.What is the smallest number you must add to 50 to make it divisible by 3? (a) 1 (b) 2
19. 19.Which of these numbers is divisible by 6? (a) 1783 (b) 1777 (c) 1771 (d) 1776
20. 20.What is the smallest number you must add to 221 to make it divisible by 10? (a) 8 (b) 7 (c) 1 (d) 9
21. 21.Which of these numbers is divisible by 6? (a) 5070 (b) 5073 (c) 5081 (d) 5068
22. 22.What is the remainder when 463 is divided by 11? (a) 10 (b) 2 (c) 1 (d) 0
23. 23.Which of these numbers is divisible by 8? (a) 2668 (b) 2655 (c) 2663 (d) 2656
24. 24.Which of these numbers is divisible by 9? (a) 7335 (b) 7333 (c) 7347 (d) 7342
25. 25.What is the remainder when 175 is divided by 6? (a) 2 (b) 0 (c) 1 (d) 5
26. 26.What is the remainder when 181 is divided by 7? (a) 1 (b) 5 (c) 6 (d) 4
27. 27.Which of these numbers is divisible by 4? (a) 494 (b) 492 (c) 499 (d) 490
28. 28.What is the remainder when 259 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
29. 29.What is the smallest number you must add to 628 to make it divisible by 10? (a) 3 (b) 1 (c) 8 (d) 2
30. 30.What is the remainder when 67 is divided by 3? (a) 2 (b) 1 (c) 0
31. 31.What is the smallest number you must add to 430 to make it divisible by 6? (a) 1 (b) 2 (c) 4 (d) 3
32. 32.Which of these numbers is divisible by 4? (a) 1468 (b) 1466 (c) 1470 (d) 1473
33. 33.What is the units digit of 4^15? (a) 5 (b) 6 (c) 3 (d) 4
34. 34.What is the remainder when 127 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
35. 35.Which of these numbers is divisible by 11? (a) 7906 (b) 7893 (c) 7887 (d) 7886
36. 36.What is the remainder when 116 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 1
37. 37.What is the units digit of 2^22? (a) 4 (b) 6 (c) 2 (d) 8
38. 38.What is the smallest number you must add to 693 to make it divisible by 10? (a) 6 (b) 7 (c) 8 (d) 3
39. 39.Which of these numbers is divisible by 11? (a) 6171 (b) 6180 (c) 6162 (d) 6184
40. 40.Which of these numbers is divisible by 4? (a) 3207 (b) 3201 (c) 3211 (d) 3204

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