Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 421 · 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 2^28? (a) 4 (b) 8 (c) 6 (d) 2
2. 2.What is the smallest number you must add to 150 to make it divisible by 4? (a) 3 (b) 1 (c) 2
3. 3.What is the remainder when 418 is divided by 10? (a) 2 (b) 8 (c) 9 (d) 7
4. 4.Which of these numbers is divisible by 6? (a) 2809 (b) 2802 (c) 2806 (d) 2800
5. 5.What is the smallest number you must add to 555 to make it divisible by 12? (a) 9 (b) 8 (c) 3 (d) 10
6. 6.What is the units digit of 8^16? (a) 6 (b) 4 (c) 2 (d) 8
7. 7.Which of these numbers is divisible by 9? (a) 4483 (b) 4476 (c) 4473 (d) 4467
8. 8.What is the units digit of 7^44? (a) 9 (b) 1 (c) 7 (d) 3
9. 9.Which of these numbers is divisible by 8? (a) 5892 (b) 5900 (c) 5905 (d) 5896
10. 10.Which of these numbers is divisible by 3? (a) 919 (b) 915 (c) 913 (d) 917
11. 11.What is the smallest number you must add to 93 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 4
12. 12.What is the remainder when 482 is divided by 6? (a) 1 (b) 2 (c) 4 (d) 3
13. 13.Which of these numbers is divisible by 11? (a) 7955 (b) 7939 (c) 7942 (d) 7943
14. 14.What is the units digit of 2^22? (a) 2 (b) 8 (c) 4 (d) 6
15. 15.What is the smallest number you must add to 857 to make it divisible by 11? (a) 1 (b) 10 (c) 3 (d) 2
16. 16.What is the remainder when 89 is divided by 3? (a) 0 (b) 1 (c) 2
17. 17.What is the remainder when 304 is divided by 9? (a) 8 (b) 2 (c) 7 (d) 6
18. 18.What is the remainder when 488 is divided by 10? (a) 2 (b) 7 (c) 8 (d) 9
19. 19.Which of these numbers is divisible by 3? (a) 1321 (b) 1324 (c) 1318 (d) 1320
20. 20.What is the remainder when 298 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
21. 21.What is the remainder when 696 is divided by 9? (a) 6 (b) 2 (c) 3 (d) 4
22. 22.Which of these numbers is divisible by 6? (a) 4774 (b) 4765 (c) 4759 (d) 4764
23. 23.What is the smallest number you must add to 635 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
24. 24.What is the smallest number you must add to 628 to make it divisible by 10? (a) 1 (b) 8 (c) 3 (d) 2
25. 25.What is the units digit of 2^14? (a) 8 (b) 6 (c) 4 (d) 2
26. 26.What is the units digit of 8^23? (a) 8 (b) 6 (c) 2 (d) 4
27. 27.Which of these numbers is divisible by 6? (a) 3269 (b) 3274 (c) 3263 (d) 3264
28. 28.What is the remainder when 329 is divided by 8? (a) 1 (b) 7 (c) 0 (d) 2
29. 29.What is the remainder when 113 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
30. 30.Which of these numbers is divisible by 6? (a) 380 (b) 387 (c) 375 (d) 378
31. 31.Which of these numbers is divisible by 9? (a) 5630 (b) 5636 (c) 5620 (d) 5625
32. 32.What is the smallest number you must add to 205 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 5
33. 33.What is the smallest number you must add to 352 to make it divisible by 7? (a) 4 (b) 5 (c) 2 (d) 6
34. 34.What is the smallest number you must add to 69 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
35. 35.Which of these numbers is divisible by 11? (a) 1894 (b) 1919 (c) 1903 (d) 1911
36. 36.What is the units digit of 2^21? (a) 4 (b) 8 (c) 6 (d) 2
37. 37.What is the remainder when 104 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
38. 38.Which of these numbers is divisible by 9? (a) 6834 (b) 6848 (c) 6840 (d) 6857
39. 39.Which of these numbers is divisible by 4? (a) 895 (b) 888 (c) 890 (d) 886
40. 40.What is the smallest number you must add to 98 to make it divisible by 4? (a) 1 (b) 2 (c) 3

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