Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 213 · 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 131 is divided by 3? (a) 2 (b) 1 (c) 0
2. 2.Which of these numbers is divisible by 8? (a) 6724 (b) 6728 (c) 6740 (d) 6734
3. 3.What is the units digit of 2^32? (a) 4 (b) 8 (c) 6 (d) 2
4. 4.Which of these numbers is divisible by 3? (a) 232 (b) 235 (c) 229 (d) 231
5. 5.Which of these numbers is divisible by 9? (a) 6740 (b) 6723 (c) 6720 (d) 6726
6. 6.What is the smallest number you must add to 245 to make it divisible by 12? (a) 8 (b) 7 (c) 5 (d) 6
7. 7.What is the smallest number you must add to 224 to make it divisible by 3? (a) 2 (b) 1
8. 8.Which of these numbers is divisible by 8? (a) 5652 (b) 5661 (c) 5656 (d) 5665
9. 9.What is the units digit of 8^48? (a) 6 (b) 8 (c) 4 (d) 2
10. 10.What is the remainder when 85 is divided by 3? (a) 2 (b) 1 (c) 0
11. 11.Which of these numbers is divisible by 11? (a) 1475 (b) 1487 (c) 1474 (d) 1470
12. 12.What is the smallest number you must add to 181 to make it divisible by 10? (a) 1 (b) 8 (c) 7 (d) 9
13. 13.What is the units digit of 4^52? (a) 4 (b) 7 (c) 5 (d) 6
14. 14.What is the smallest number you must add to 771 to make it divisible by 12? (a) 3 (b) 10 (c) 8 (d) 9
15. 15.What is the units digit of 3^22? (a) 3 (b) 9 (c) 1 (d) 7
16. 16.Which of these numbers is divisible by 11? (a) 1353 (b) 1345 (c) 1366 (d) 1356
17. 17.What is the remainder when 316 is divided by 5? (a) 1 (b) 0 (c) 2 (d) 4
18. 18.Which of these numbers is divisible by 4? (a) 2597 (b) 2592 (c) 2593 (d) 2591
19. 19.Which of these numbers is divisible by 4? (a) 2630 (b) 2632 (c) 2635 (d) 2639
20. 20.What is the remainder when 287 is divided by 8? (a) 5 (b) 1 (c) 6 (d) 7
21. 21.Which of these numbers is divisible by 11? (a) 3286 (b) 3276 (c) 3298 (d) 3278
22. 22.Which of these numbers is divisible by 4? (a) 2190 (b) 2187 (c) 2195 (d) 2188
23. 23.What is the remainder when 577 is divided by 8? (a) 1 (b) 7 (c) 2 (d) 0
24. 24.What is the units digit of 8^10? (a) 6 (b) 2 (c) 4 (d) 8
25. 25.Which of these numbers is divisible by 4? (a) 262 (b) 257 (c) 255 (d) 256
26. 26.What is the smallest number you must add to 145 to make it divisible by 9? (a) 8 (b) 6 (c) 7 (d) 1
27. 27.Which of these numbers is divisible by 9? (a) 3092 (b) 3112 (c) 3096 (d) 3097
28. 28.Which of these numbers is divisible by 11? (a) 4178 (b) 4164 (c) 4169 (d) 4185
29. 29.Which of these numbers is divisible by 11? (a) 3798 (b) 3795 (c) 3816 (d) 3785
30. 30.Which of these numbers is divisible by 8? (a) 6787 (b) 6793 (c) 6801 (d) 6792
31. 31.Which of these numbers is divisible by 9? (a) 2548 (b) 2539 (c) 2559 (d) 2547
32. 32.What is the remainder when 218 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
33. 33.What is the smallest number you must add to 275 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 6
34. 34.What is the units digit of 9^42? (a) 2 (b) 1 (c) 0 (d) 9
35. 35.What is the remainder when 146 is divided by 3? (a) 1 (b) 2 (c) 0
36. 36.Which of these numbers is divisible by 11? (a) 8061 (b) 8043 (c) 8065 (d) 8052
37. 37.What is the smallest number you must add to 226 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
38. 38.What is the remainder when 815 is divided by 12? (a) 10 (b) 1 (c) 11 (d) 9
39. 39.What is the smallest number you must add to 883 to make it divisible by 11? (a) 9 (b) 7 (c) 3 (d) 8
40. 40.What is the smallest number you must add to 525 to make it divisible by 8? (a) 5 (b) 2 (c) 4 (d) 3

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