Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 184 · 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 4^26? (a) 5 (b) 4 (c) 7 (d) 6
2. 2.What is the remainder when 137 is divided by 12? (a) 6 (b) 4 (c) 5 (d) 7
3. 3.What is the smallest number you must add to 159 to make it divisible by 4? (a) 2 (b) 3 (c) 1
4. 4.What is the smallest number you must add to 301 to make it divisible by 8? (a) 2 (b) 3 (c) 4 (d) 5
5. 5.What is the smallest number you must add to 185 to make it divisible by 9? (a) 3 (b) 5 (c) 6 (d) 4
6. 6.What is the smallest number you must add to 381 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 3
7. 7.What is the smallest number you must add to 509 to make it divisible by 9? (a) 4 (b) 3 (c) 6 (d) 5
8. 8.What is the units digit of 7^58? (a) 7 (b) 9 (c) 1 (d) 3
9. 9.Which of these numbers is divisible by 6? (a) 4141 (b) 4149 (c) 4146 (d) 4156
10. 10.What is the remainder when 737 is divided by 10? (a) 3 (b) 8 (c) 6 (d) 7
11. 11.Which of these numbers is divisible by 11? (a) 3823 (b) 3829 (c) 3846 (d) 3828
12. 12.Which of these numbers is divisible by 4? (a) 781 (b) 773 (c) 776 (d) 777
13. 13.What is the remainder when 229 is divided by 7? (a) 2 (b) 6 (c) 4 (d) 5
14. 14.What is the remainder when 93 is divided by 7? (a) 3 (b) 5 (c) 1 (d) 2
15. 15.Which of these numbers is divisible by 11? (a) 2621 (b) 2636 (c) 2618 (d) 2617
16. 16.What is the smallest number you must add to 402 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
17. 17.Which of these numbers is divisible by 3? (a) 1706 (b) 1707 (c) 1709 (d) 1711
18. 18.Which of these numbers is divisible by 8? (a) 769 (b) 754 (c) 760 (d) 762
19. 19.Which of these numbers is divisible by 8? (a) 3056 (b) 3053 (c) 3059 (d) 3069
20. 20.What is the smallest number you must add to 448 to make it divisible by 11? (a) 2 (b) 8 (c) 4 (d) 3
21. 21.What is the remainder when 89 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
22. 22.Which of these numbers is divisible by 6? (a) 1398 (b) 1393 (c) 1407 (d) 1402
23. 23.What is the units digit of 2^55? (a) 2 (b) 8 (c) 6 (d) 4
24. 24.What is the units digit of 3^23? (a) 9 (b) 7 (c) 1 (d) 3
25. 25.Which of these numbers is divisible by 11? (a) 900 (b) 902 (c) 909 (d) 922
26. 26.What is the remainder when 322 is divided by 8? (a) 3 (b) 1 (c) 2 (d) 6
27. 27.What is the units digit of 7^42? (a) 7 (b) 3 (c) 9 (d) 1
28. 28.What is the smallest number you must add to 663 to make it divisible by 9? (a) 6 (b) 2 (c) 4 (d) 3
29. 29.Which of these numbers is divisible by 8? (a) 5734 (b) 5740 (c) 5736 (d) 5745
30. 30.What is the remainder when 145 is divided by 12? (a) 1 (b) 2 (c) 11 (d) 0
31. 31.Which of these numbers is divisible by 9? (a) 5346 (b) 5354 (c) 5338 (d) 5358
32. 32.What is the smallest number you must add to 241 to make it divisible by 10? (a) 8 (b) 1 (c) 7 (d) 9
33. 33.Which of these numbers is divisible by 8? (a) 344 (b) 353 (c) 341 (d) 346
34. 34.What is the units digit of 7^31? (a) 9 (b) 1 (c) 3 (d) 7
35. 35.What is the remainder when 89 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
36. 36.What is the units digit of 9^24? (a) 2 (b) 0 (c) 1 (d) 9
37. 37.What is the smallest number you must add to 391 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
38. 38.What is the remainder when 382 is divided by 9? (a) 5 (b) 4 (c) 6 (d) 3
39. 39.What is the remainder when 266 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
40. 40.What is the remainder when 86 is divided by 8? (a) 5 (b) 6 (c) 7 (d) 2

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