Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 433 · 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 smallest number you must add to 782 to make it divisible by 10? (a) 2 (b) 8 (c) 7 (d) 9
2. 2.What is the remainder when 389 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 1
3. 3.What is the remainder when 86 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 2
4. 4.What is the units digit of 8^22? (a) 2 (b) 8 (c) 4 (d) 6
5. 5.Which of these numbers is divisible by 8? (a) 1000 (b) 1005 (c) 1014 (d) 997
6. 6.What is the remainder when 172 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
7. 7.Which of these numbers is divisible by 4? (a) 2831 (b) 2833 (c) 2828 (d) 2826
8. 8.Which of these numbers is divisible by 6? (a) 3311 (b) 3317 (c) 3306 (d) 3304
9. 9.What is the units digit of 7^47? (a) 3 (b) 9 (c) 7 (d) 1
10. 10.What is the units digit of 4^37? (a) 6 (b) 3 (c) 4 (d) 5
11. 11.Which of these numbers is divisible by 3? (a) 983 (b) 979 (c) 981 (d) 985
12. 12.What is the smallest number you must add to 131 to make it divisible by 7? (a) 3 (b) 5 (c) 1 (d) 2
13. 13.Which of these numbers is divisible by 8? (a) 6924 (b) 6920 (c) 6935 (d) 6914
14. 14.What is the units digit of 2^42? (a) 2 (b) 6 (c) 4 (d) 8
15. 15.What is the units digit of 4^14? (a) 4 (b) 6 (c) 7 (d) 5
16. 16.What is the units digit of 8^16? (a) 4 (b) 2 (c) 8 (d) 6
17. 17.What is the remainder when 719 is divided by 10? (a) 9 (b) 7 (c) 8 (d) 1
18. 18.What is the remainder when 288 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
19. 19.Which of these numbers is divisible by 11? (a) 8644 (b) 8635 (c) 8633 (d) 8651
20. 20.What is the smallest number you must add to 177 to make it divisible by 10? (a) 7 (b) 2 (c) 3 (d) 4
21. 21.Which of these numbers is divisible by 9? (a) 6620 (b) 6614 (c) 6627 (d) 6615
22. 22.What is the units digit of 8^57? (a) 8 (b) 6 (c) 2 (d) 4
23. 23.What is the smallest number you must add to 235 to make it divisible by 4? (a) 1 (b) 3 (c) 2
24. 24.What is the remainder when 181 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
25. 25.What is the remainder when 229 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
26. 26.What is the remainder when 319 is divided by 9? (a) 4 (b) 5 (c) 3 (d) 6
27. 27.What is the smallest number you must add to 441 to make it divisible by 11? (a) 9 (b) 8 (c) 1 (d) 10
28. 28.What is the units digit of 4^42? (a) 4 (b) 6 (c) 5 (d) 7
29. 29.What is the smallest number you must add to 105 to make it divisible by 4? (a) 1 (b) 2 (c) 3
30. 30.What is the smallest number you must add to 357 to make it divisible by 12? (a) 2 (b) 9 (c) 4 (d) 3
31. 31.Which of these numbers is divisible by 3? (a) 1769 (b) 1775 (c) 1770 (d) 1772
32. 32.What is the units digit of 7^34? (a) 3 (b) 7 (c) 1 (d) 9
33. 33.Which of these numbers is divisible by 8? (a) 3256 (b) 3262 (c) 3266 (d) 3251
34. 34.What is the smallest number you must add to 256 to make it divisible by 12? (a) 8 (b) 4 (c) 7 (d) 9
35. 35.What is the units digit of 4^20? (a) 5 (b) 4 (c) 7 (d) 6
36. 36.Which of these numbers is divisible by 6? (a) 4876 (b) 4878 (c) 4879 (d) 4887
37. 37.What is the units digit of 9^58? (a) 9 (b) 0 (c) 2 (d) 1
38. 38.Which of these numbers is divisible by 8? (a) 1547 (b) 1540 (c) 1531 (d) 1536
39. 39.Which of these numbers is divisible by 11? (a) 7997 (b) 8010 (c) 7992 (d) 8003
40. 40.What is the remainder when 430 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7

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