Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 239 · 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.Which of these numbers is divisible by 6? (a) 4397 (b) 4388 (c) 4382 (d) 4386
2. 2.What is the remainder when 751 is divided by 12? (a) 6 (b) 7 (c) 8 (d) 5
3. 3.What is the units digit of 9^45? (a) 1 (b) 7 (c) 8 (d) 9
4. 4.What is the smallest number you must add to 193 to make it divisible by 3? (a) 2 (b) 1
5. 5.Which of these numbers is divisible by 11? (a) 1511 (b) 1488 (c) 1499 (d) 1496
6. 6.What is the units digit of 2^31? (a) 4 (b) 2 (c) 6 (d) 8
7. 7.What is the remainder when 263 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
8. 8.What is the remainder when 194 is divided by 10? (a) 6 (b) 5 (c) 3 (d) 4
9. 9.What is the units digit of 2^27? (a) 8 (b) 4 (c) 6 (d) 2
10. 10.Which of these numbers is divisible by 9? (a) 3900 (b) 3886 (c) 3888 (d) 3890
11. 11.What is the smallest number you must add to 788 to make it divisible by 11? (a) 7 (b) 3 (c) 4 (d) 5
12. 12.What is the units digit of 4^38? (a) 4 (b) 7 (c) 5 (d) 6
13. 13.Which of these numbers is divisible by 8? (a) 4280 (b) 4284 (c) 4289 (d) 4277
14. 14.What is the smallest number you must add to 158 to make it divisible by 3? (a) 2 (b) 1
15. 15.Which of these numbers is divisible by 9? (a) 6948 (b) 6952 (c) 6958 (d) 6942
16. 16.What is the units digit of 4^21? (a) 5 (b) 4 (c) 6 (d) 3
17. 17.What is the smallest number you must add to 237 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
18. 18.What is the smallest number you must add to 181 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
19. 19.What is the units digit of 4^55? (a) 3 (b) 4 (c) 6 (d) 5
20. 20.What is the units digit of 9^30? (a) 0 (b) 1 (c) 2 (d) 9
21. 21.Which of these numbers is divisible by 8? (a) 5806 (b) 5795 (c) 5814 (d) 5800
22. 22.What is the units digit of 3^38? (a) 1 (b) 7 (c) 9 (d) 3
23. 23.What is the smallest number you must add to 61 to make it divisible by 3? (a) 1 (b) 2
24. 24.Which of these numbers is divisible by 8? (a) 914 (b) 912 (c) 924 (d) 911
25. 25.What is the smallest number you must add to 396 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
26. 26.What is the remainder when 314 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 1
27. 27.Which of these numbers is divisible by 9? (a) 2174 (b) 2191 (c) 2181 (d) 2178
28. 28.What is the smallest number you must add to 786 to make it divisible by 11? (a) 8 (b) 7 (c) 6 (d) 5
29. 29.What is the smallest number you must add to 81 to make it divisible by 4? (a) 1 (b) 3 (c) 2
30. 30.What is the smallest number you must add to 331 to make it divisible by 6? (a) 4 (b) 5 (c) 1 (d) 3
31. 31.What is the remainder when 491 is divided by 12? (a) 11 (b) 9 (c) 1 (d) 10
32. 32.Which of these numbers is divisible by 8? (a) 2064 (b) 2065 (c) 2079 (d) 2059
33. 33.Which of these numbers is divisible by 3? (a) 1138 (b) 1137 (c) 1141 (d) 1136
34. 34.What is the smallest number you must add to 57 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
35. 35.What is the remainder when 172 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 5
36. 36.What is the units digit of 8^30? (a) 4 (b) 6 (c) 8 (d) 2
37. 37.What is the smallest number you must add to 67 to make it divisible by 4? (a) 1 (b) 3 (c) 2
38. 38.What is the smallest number you must add to 246 to make it divisible by 12? (a) 6 (b) 7 (c) 8 (d) 5
39. 39.What is the remainder when 321 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
40. 40.What is the smallest number you must add to 496 to make it divisible by 10? (a) 6 (b) 4 (c) 5 (d) 3

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