Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 282 · 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 97 to make it divisible by 3? (a) 1 (b) 2
2. 2.What is the remainder when 161 is divided by 8? (a) 2 (b) 7 (c) 1 (d) 0
3. 3.What is the smallest number you must add to 214 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the remainder when 533 is divided by 7? (a) 1 (b) 2 (c) 6 (d) 0
5. 5.What is the remainder when 110 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
6. 6.What is the smallest number you must add to 120 to make it divisible by 9? (a) 5 (b) 6 (c) 7 (d) 3
7. 7.What is the units digit of 4^48? (a) 7 (b) 4 (c) 5 (d) 6
8. 8.What is the units digit of 9^14? (a) 9 (b) 0 (c) 1 (d) 2
9. 9.What is the smallest number you must add to 442 to make it divisible by 8? (a) 7 (b) 2 (c) 5 (d) 6
10. 10.What is the remainder when 300 is divided by 8? (a) 6 (b) 3 (c) 4 (d) 5
11. 11.What is the remainder when 136 is divided by 11? (a) 7 (b) 4 (c) 5 (d) 3
12. 12.Which of these numbers is divisible by 6? (a) 4655 (b) 4639 (c) 4644 (d) 4645
13. 13.Which of these numbers is divisible by 8? (a) 2960 (b) 2962 (c) 2973 (d) 2957
14. 14.What is the remainder when 546 is divided by 9? (a) 6 (b) 7 (c) 5 (d) 3
15. 15.What is the remainder when 467 is divided by 11? (a) 6 (b) 7 (c) 5 (d) 4
16. 16.What is the remainder when 557 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
17. 17.What is the smallest number you must add to 196 to make it divisible by 3? (a) 2 (b) 1
18. 18.What is the remainder when 263 is divided by 11? (a) 9 (b) 8 (c) 10 (d) 1
19. 19.What is the units digit of 3^35? (a) 9 (b) 3 (c) 7 (d) 1
20. 20.Which of these numbers is divisible by 11? (a) 4207 (b) 4213 (c) 4215 (d) 4228
21. 21.What is the units digit of 3^43? (a) 9 (b) 1 (c) 3 (d) 7
22. 22.What is the smallest number you must add to 438 to make it divisible by 8? (a) 2 (b) 6 (c) 3 (d) 1
23. 23.Which of these numbers is divisible by 4? (a) 2580 (b) 2577 (c) 2583 (d) 2587
24. 24.What is the smallest number you must add to 344 to make it divisible by 7? (a) 4 (b) 5 (c) 1 (d) 6
25. 25.What is the remainder when 365 is divided by 8? (a) 5 (b) 6 (c) 3 (d) 4
26. 26.What is the units digit of 7^15? (a) 7 (b) 3 (c) 1 (d) 9
27. 27.Which of these numbers is divisible by 3? (a) 2335 (b) 2331 (c) 2330 (d) 2333
28. 28.What is the remainder when 155 is divided by 7? (a) 0 (b) 6 (c) 1 (d) 2
29. 29.What is the units digit of 8^41? (a) 6 (b) 8 (c) 4 (d) 2
30. 30.Which of these numbers is divisible by 8? (a) 1933 (b) 1922 (c) 1938 (d) 1928
31. 31.Which of these numbers is divisible by 3? (a) 2604 (b) 2603 (c) 2606 (d) 2608
32. 32.What is the remainder when 155 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 2
33. 33.What is the smallest number you must add to 492 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
34. 34.Which of these numbers is divisible by 9? (a) 4410 (b) 4416 (c) 4409 (d) 4422
35. 35.Which of these numbers is divisible by 8? (a) 4087 (b) 4089 (c) 4079 (d) 4080
36. 36.Which of these numbers is divisible by 8? (a) 5079 (b) 5080 (c) 5095 (d) 5084
37. 37.What is the remainder when 481 is divided by 11? (a) 8 (b) 3 (c) 7 (d) 9
38. 38.What is the remainder when 299 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
39. 39.What is the smallest number you must add to 87 to make it divisible by 6? (a) 5 (b) 4 (c) 2 (d) 3
40. 40.What is the units digit of 7^60? (a) 9 (b) 3 (c) 7 (d) 1

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