Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 75 · 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 7^31? (a) 1 (b) 7 (c) 3 (d) 9
2. 2.Which of these numbers is divisible by 4? (a) 188 (b) 190 (c) 194 (d) 185
3. 3.What is the remainder when 188 is divided by 12? (a) 4 (b) 7 (c) 8 (d) 9
4. 4.Which of these numbers is divisible by 11? (a) 616 (b) 635 (c) 622 (d) 615
5. 5.Which of these numbers is divisible by 9? (a) 7655 (b) 7641 (c) 7648 (d) 7638
6. 6.What is the remainder when 341 is divided by 8? (a) 6 (b) 3 (c) 5 (d) 4
7. 7.Which of these numbers is divisible by 9? (a) 5361 (b) 5350 (c) 5355 (d) 5370
8. 8.What is the remainder when 118 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
9. 9.What is the smallest number you must add to 350 to make it divisible by 11? (a) 2 (b) 9 (c) 3 (d) 1
10. 10.What is the smallest number you must add to 463 to make it divisible by 9? (a) 4 (b) 5 (c) 6 (d) 7
11. 11.Which of these numbers is divisible by 3? (a) 2649 (b) 2650 (c) 2653 (d) 2647
12. 12.What is the smallest number you must add to 218 to make it divisible by 10? (a) 2 (b) 8 (c) 3 (d) 1
13. 13.Which of these numbers is divisible by 3? (a) 870 (b) 875 (c) 871 (d) 868
14. 14.What is the remainder when 502 is divided by 10? (a) 1 (b) 3 (c) 2 (d) 8
15. 15.What is the units digit of 2^18? (a) 4 (b) 6 (c) 8 (d) 2
16. 16.What is the units digit of 2^43? (a) 4 (b) 2 (c) 8 (d) 6
17. 17.What is the remainder when 157 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
18. 18.What is the smallest number you must add to 183 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 5
19. 19.What is the units digit of 4^32? (a) 4 (b) 7 (c) 5 (d) 6
20. 20.What is the units digit of 8^16? (a) 2 (b) 8 (c) 6 (d) 4
21. 21.What is the remainder when 49 is divided by 3? (a) 2 (b) 0 (c) 1
22. 22.What is the units digit of 4^25? (a) 5 (b) 4 (c) 6 (d) 3
23. 23.Which of these numbers is divisible by 6? (a) 4130 (b) 4123 (c) 4128 (d) 4135
24. 24.What is the units digit of 7^15? (a) 1 (b) 7 (c) 3 (d) 9
25. 25.What is the remainder when 534 is divided by 7? (a) 2 (b) 5 (c) 1 (d) 3
26. 26.Which of these numbers is divisible by 3? (a) 1656 (b) 1657 (c) 1661 (d) 1654
27. 27.What is the smallest number you must add to 82 to make it divisible by 3? (a) 2 (b) 1
28. 28.What is the remainder when 680 is divided by 11? (a) 10 (b) 8 (c) 2 (d) 9
29. 29.What is the units digit of 3^41? (a) 9 (b) 3 (c) 7 (d) 1
30. 30.What is the smallest number you must add to 173 to make it divisible by 4? (a) 1 (b) 3 (c) 2
31. 31.Which of these numbers is divisible by 11? (a) 5047 (b) 5049 (c) 5068 (d) 5057
32. 32.What is the remainder when 136 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
33. 33.Which of these numbers is divisible by 11? (a) 6310 (b) 6285 (c) 6292 (d) 6295
34. 34.Which of these numbers is divisible by 4? (a) 2251 (b) 2243 (c) 2244 (d) 2245
35. 35.Which of these numbers is divisible by 11? (a) 1063 (b) 1055 (c) 1056 (d) 1073
36. 36.Which of these numbers is divisible by 6? (a) 4665 (b) 4653 (c) 4659 (d) 4656
37. 37.What is the remainder when 283 is divided by 10? (a) 7 (b) 3 (c) 4 (d) 2
38. 38.What is the smallest number you must add to 224 to make it divisible by 11? (a) 4 (b) 6 (c) 7 (d) 8
39. 39.What is the units digit of 2^29? (a) 6 (b) 8 (c) 2 (d) 4
40. 40.Which of these numbers is divisible by 6? (a) 2844 (b) 2853 (c) 2840 (d) 2848

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad