Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 254 · 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 remainder when 675 is divided by 12? (a) 2 (b) 9 (c) 4 (d) 3
2. 2.What is the smallest number you must add to 554 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
3. 3.What is the remainder when 422 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
4. 4.What is the units digit of 2^42? (a) 4 (b) 6 (c) 2 (d) 8
5. 5.What is the smallest number you must add to 114 to make it divisible by 8? (a) 7 (b) 5 (c) 2 (d) 6
6. 6.Which of these numbers is divisible by 9? (a) 4269 (b) 4264 (c) 4266 (d) 4279
7. 7.Which of these numbers is divisible by 6? (a) 1855 (b) 1848 (c) 1853 (d) 1846
8. 8.Which of these numbers is divisible by 9? (a) 1148 (b) 1141 (c) 1159 (d) 1143
9. 9.What is the units digit of 2^39? (a) 2 (b) 6 (c) 4 (d) 8
10. 10.What is the smallest number you must add to 172 to make it divisible by 8? (a) 4 (b) 6 (c) 3 (d) 5
11. 11.What is the smallest number you must add to 331 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
12. 12.What is the remainder when 61 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
13. 13.What is the remainder when 729 is divided by 10? (a) 7 (b) 8 (c) 1 (d) 9
14. 14.What is the remainder when 620 is divided by 11? (a) 5 (b) 7 (c) 3 (d) 4
15. 15.What is the smallest number you must add to 612 to make it divisible by 11? (a) 5 (b) 4 (c) 3 (d) 7
16. 16.What is the smallest number you must add to 251 to make it divisible by 8? (a) 6 (b) 4 (c) 5 (d) 3
17. 17.What is the units digit of 2^26? (a) 2 (b) 8 (c) 4 (d) 6
18. 18.Which of these numbers is divisible by 4? (a) 3446 (b) 3450 (c) 3453 (d) 3448
19. 19.What is the remainder when 647 is divided by 10? (a) 8 (b) 3 (c) 7 (d) 6
20. 20.What is the units digit of 7^21? (a) 7 (b) 3 (c) 1 (d) 9
21. 21.Which of these numbers is divisible by 11? (a) 1940 (b) 1967 (c) 1947 (d) 1955
22. 22.What is the remainder when 261 is divided by 7? (a) 2 (b) 5 (c) 1 (d) 3
23. 23.What is the remainder when 349 is divided by 6? (a) 5 (b) 2 (c) 1 (d) 0
24. 24.What is the units digit of 8^32? (a) 6 (b) 8 (c) 2 (d) 4
25. 25.What is the units digit of 8^15? (a) 4 (b) 2 (c) 6 (d) 8
26. 26.Which of these numbers is divisible by 9? (a) 2636 (b) 2615 (c) 2619 (d) 2621
27. 27.Which of these numbers is divisible by 9? (a) 3968 (b) 3954 (c) 3974 (d) 3960
28. 28.What is the units digit of 8^45? (a) 4 (b) 8 (c) 2 (d) 6
29. 29.What is the remainder when 44 is divided by 3? (a) 2 (b) 1 (c) 0
30. 30.Which of these numbers is divisible by 8? (a) 3224 (b) 3231 (c) 3222 (d) 3239
31. 31.Which of these numbers is divisible by 8? (a) 5594 (b) 5578 (c) 5584 (d) 5587
32. 32.What is the remainder when 808 is divided by 12? (a) 8 (b) 5 (c) 4 (d) 3
33. 33.Which of these numbers is divisible by 8? (a) 1703 (b) 1695 (c) 1696 (d) 1710
34. 34.What is the units digit of 2^33? (a) 8 (b) 2 (c) 4 (d) 6
35. 35.What is the remainder when 78 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
36. 36.What is the remainder when 267 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
37. 37.What is the smallest number you must add to 515 to make it divisible by 8? (a) 3 (b) 5 (c) 6 (d) 4
38. 38.Which of these numbers is divisible by 4? (a) 1625 (b) 1621 (c) 1620 (d) 1617
39. 39.What is the units digit of 2^36? (a) 4 (b) 6 (c) 8 (d) 2
40. 40.What is the smallest number you must add to 443 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 2

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