Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 581 · 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 206 to make it divisible by 4? (a) 2 (b) 1 (c) 3
2. 2.Which of these numbers is divisible by 3? (a) 2020 (b) 2017 (c) 2016 (d) 2014
3. 3.What is the units digit of 8^22? (a) 4 (b) 2 (c) 6 (d) 8
4. 4.What is the units digit of 8^54? (a) 6 (b) 2 (c) 8 (d) 4
5. 5.What is the remainder when 156 is divided by 11? (a) 2 (b) 9 (c) 1 (d) 3
6. 6.Which of these numbers is divisible by 4? (a) 2931 (b) 2924 (c) 2923 (d) 2926
7. 7.What is the remainder when 295 is divided by 7? (a) 0 (b) 1 (c) 6 (d) 2
8. 8.What is the remainder when 128 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
9. 9.What is the units digit of 8^41? (a) 2 (b) 8 (c) 6 (d) 4
10. 10.What is the smallest number you must add to 179 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the smallest number you must add to 505 to make it divisible by 9? (a) 1 (b) 6 (c) 8 (d) 7
12. 12.What is the remainder when 594 is divided by 8? (a) 6 (b) 2 (c) 3 (d) 1
13. 13.What is the units digit of 8^50? (a) 2 (b) 6 (c) 8 (d) 4
14. 14.What is the smallest number you must add to 362 to make it divisible by 8? (a) 5 (b) 2 (c) 7 (d) 6
15. 15.Which of these numbers is divisible by 4? (a) 424 (b) 427 (c) 422 (d) 431
16. 16.Which of these numbers is divisible by 3? (a) 1445 (b) 1443 (c) 1442 (d) 1447
17. 17.What is the smallest number you must add to 916 to make it divisible by 12? (a) 9 (b) 7 (c) 4 (d) 8
18. 18.What is the smallest number you must add to 505 to make it divisible by 8? (a) 1 (b) 5 (c) 6 (d) 7
19. 19.Which of these numbers is divisible by 8? (a) 5996 (b) 5986 (c) 5992 (d) 6001
20. 20.What is the units digit of 8^15? (a) 4 (b) 8 (c) 2 (d) 6
21. 21.What is the remainder when 147 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
22. 22.What is the units digit of 2^41? (a) 8 (b) 6 (c) 2 (d) 4
23. 23.What is the remainder when 49 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
24. 24.What is the smallest number you must add to 400 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 4
25. 25.Which of these numbers is divisible by 6? (a) 2579 (b) 2563 (c) 2568 (d) 2571
26. 26.What is the smallest number you must add to 170 to make it divisible by 12? (a) 10 (b) 11 (c) 9 (d) 2
27. 27.Which of these numbers is divisible by 3? (a) 1153 (b) 1149 (c) 1148 (d) 1150
28. 28.Which of these numbers is divisible by 9? (a) 7458 (b) 7443 (c) 7448 (d) 7438
29. 29.What is the units digit of 8^17? (a) 2 (b) 4 (c) 8 (d) 6
30. 30.What is the smallest number you must add to 162 to make it divisible by 4? (a) 2 (b) 1 (c) 3
31. 31.Which of these numbers is divisible by 11? (a) 7911 (b) 7903 (c) 7897 (d) 7898
32. 32.What is the units digit of 7^25? (a) 7 (b) 1 (c) 9 (d) 3
33. 33.Which of these numbers is divisible by 3? (a) 445 (b) 448 (c) 443 (d) 444
34. 34.What is the units digit of 8^16? (a) 2 (b) 6 (c) 4 (d) 8
35. 35.What is the smallest number you must add to 103 to make it divisible by 4? (a) 2 (b) 1 (c) 3
36. 36.Which of these numbers is divisible by 4? (a) 3390 (b) 3388 (c) 3385 (d) 3395
37. 37.Which of these numbers is divisible by 8? (a) 1845 (b) 1851 (c) 1838 (d) 1840
38. 38.What is the remainder when 349 is divided by 9? (a) 6 (b) 7 (c) 2 (d) 8
39. 39.Which of these numbers is divisible by 6? (a) 1933 (b) 1938 (c) 1939 (d) 1949
40. 40.What is the units digit of 3^46? (a) 3 (b) 7 (c) 9 (d) 1

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