Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 866 · 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 9^13? (a) 9 (b) 7 (c) 1 (d) 8
2. 2.What is the smallest number you must add to 53 to make it divisible by 3? (a) 2 (b) 1
3. 3.Which of these numbers is divisible by 8? (a) 5475 (b) 5464 (c) 5457 (d) 5469
4. 4.What is the units digit of 3^17? (a) 1 (b) 9 (c) 7 (d) 3
5. 5.Which of these numbers is divisible by 11? (a) 9900 (b) 9892 (c) 9917 (d) 9910
6. 6.Which of these numbers is divisible by 3? (a) 1052 (b) 1045 (c) 1047 (d) 1049
7. 7.What is the units digit of 3^33? (a) 9 (b) 1 (c) 3 (d) 7
8. 8.What is the remainder when 587 is divided by 10? (a) 6 (b) 3 (c) 8 (d) 7
9. 9.What is the remainder when 294 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
10. 10.Which of these numbers is divisible by 3? (a) 1263 (b) 1261 (c) 1264 (d) 1267
11. 11.Which of these numbers is divisible by 6? (a) 2327 (b) 2329 (c) 2320 (d) 2322
12. 12.What is the remainder when 788 is divided by 10? (a) 7 (b) 2 (c) 8 (d) 9
13. 13.What is the remainder when 223 is divided by 8? (a) 6 (b) 1 (c) 5 (d) 7
14. 14.Which of these numbers is divisible by 4? (a) 1717 (b) 1710 (c) 1712 (d) 1713
15. 15.What is the smallest number you must add to 103 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 4
16. 16.Which of these numbers is divisible by 6? (a) 3576 (b) 3574 (c) 3581 (d) 3585
17. 17.What is the remainder when 88 is divided by 3? (a) 2 (b) 0 (c) 1
18. 18.What is the smallest number you must add to 483 to make it divisible by 12? (a) 3 (b) 8 (c) 10 (d) 9
19. 19.What is the units digit of 3^23? (a) 3 (b) 9 (c) 7 (d) 1
20. 20.What is the units digit of 9^23? (a) 1 (b) 9 (c) 8 (d) 7
21. 21.What is the remainder when 317 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 1
22. 22.Which of these numbers is divisible by 3? (a) 1759 (b) 1766 (c) 1763 (d) 1761
23. 23.What is the remainder when 807 is divided by 12? (a) 9 (b) 2 (c) 3 (d) 4
24. 24.Which of these numbers is divisible by 9? (a) 6703 (b) 6688 (c) 6711 (d) 6696
25. 25.What is the smallest number you must add to 430 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
26. 26.What is the units digit of 9^12? (a) 2 (b) 0 (c) 9 (d) 1
27. 27.Which of these numbers is divisible by 11? (a) 4310 (b) 4298 (c) 4281 (d) 4290
28. 28.What is the smallest number you must add to 157 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 1
29. 29.What is the smallest number you must add to 551 to make it divisible by 9? (a) 8 (b) 6 (c) 7 (d) 2
30. 30.What is the units digit of 2^25? (a) 4 (b) 6 (c) 2 (d) 8
31. 31.Which of these numbers is divisible by 4? (a) 3057 (b) 3053 (c) 3056 (d) 3061
32. 32.Which of these numbers is divisible by 3? (a) 634 (b) 632 (c) 637 (d) 633
33. 33.What is the remainder when 438 is divided by 9? (a) 5 (b) 6 (c) 7 (d) 3
34. 34.What is the smallest number you must add to 106 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
35. 35.Which of these numbers is divisible by 3? (a) 684 (b) 682 (c) 685 (d) 688
36. 36.What is the remainder when 51 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
37. 37.What is the units digit of 9^54? (a) 2 (b) 9 (c) 0 (d) 1
38. 38.What is the remainder when 130 is divided by 9? (a) 4 (b) 6 (c) 5 (d) 3
39. 39.What is the remainder when 862 is divided by 11? (a) 3 (b) 4 (c) 5 (d) 7
40. 40.Which of these numbers is divisible by 9? (a) 6444 (b) 6441 (c) 6446 (d) 6458

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