Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 432 · 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 267 to make it divisible by 4? (a) 2 (b) 1 (c) 3
2. 2.What is the remainder when 366 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
3. 3.What is the remainder when 307 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 6? (a) 3135 (b) 3124 (c) 3126 (d) 3130
5. 5.What is the remainder when 688 is divided by 9? (a) 5 (b) 3 (c) 4 (d) 6
6. 6.What is the units digit of 9^20? (a) 2 (b) 1 (c) 9 (d) 0
7. 7.What is the units digit of 9^27? (a) 7 (b) 8 (c) 9 (d) 1
8. 8.Which of these numbers is divisible by 9? (a) 414 (b) 415 (c) 425 (d) 413
9. 9.Which of these numbers is divisible by 9? (a) 3491 (b) 3494 (c) 3483 (d) 3482
10. 10.Which of these numbers is divisible by 4? (a) 3530 (b) 3524 (c) 3522 (d) 3526
11. 11.Which of these numbers is divisible by 11? (a) 2015 (b) 2010 (c) 2013 (d) 2027
12. 12.What is the units digit of 9^37? (a) 8 (b) 7 (c) 9 (d) 1
13. 13.Which of these numbers is divisible by 3? (a) 1851 (b) 1852 (c) 1849 (d) 1855
14. 14.What is the units digit of 3^15? (a) 9 (b) 1 (c) 7 (d) 3
15. 15.What is the units digit of 8^57? (a) 8 (b) 2 (c) 4 (d) 6
16. 16.What is the smallest number you must add to 373 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 2
17. 17.What is the remainder when 417 is divided by 11? (a) 1 (b) 8 (c) 9 (d) 10
18. 18.What is the remainder when 289 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
19. 19.What is the units digit of 7^17? (a) 7 (b) 9 (c) 3 (d) 1
20. 20.What is the units digit of 2^17? (a) 4 (b) 2 (c) 8 (d) 6
21. 21.Which of these numbers is divisible by 4? (a) 3082 (b) 3087 (c) 3078 (d) 3080
22. 22.What is the smallest number you must add to 181 to make it divisible by 4? (a) 1 (b) 2 (c) 3
23. 23.What is the remainder when 391 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
24. 24.Which of these numbers is divisible by 3? (a) 845 (b) 840 (c) 842 (d) 838
25. 25.Which of these numbers is divisible by 4? (a) 987 (b) 981 (c) 979 (d) 980
26. 26.What is the units digit of 9^21? (a) 1 (b) 7 (c) 8 (d) 9
27. 27.Which of these numbers is divisible by 11? (a) 860 (b) 875 (c) 857 (d) 858
28. 28.What is the units digit of 3^21? (a) 7 (b) 9 (c) 3 (d) 1
29. 29.What is the remainder when 466 is divided by 10? (a) 5 (b) 7 (c) 4 (d) 6
30. 30.What is the units digit of 8^45? (a) 8 (b) 6 (c) 4 (d) 2
31. 31.Which of these numbers is divisible by 6? (a) 4377 (b) 4382 (c) 4374 (d) 4369
32. 32.What is the remainder when 113 is divided by 7? (a) 1 (b) 0 (c) 2 (d) 6
33. 33.What is the smallest number you must add to 97 to make it divisible by 4? (a) 1 (b) 2 (c) 3
34. 34.What is the units digit of 4^46? (a) 4 (b) 6 (c) 5 (d) 7
35. 35.What is the units digit of 2^60? (a) 4 (b) 8 (c) 2 (d) 6
36. 36.What is the units digit of 8^15? (a) 4 (b) 8 (c) 6 (d) 2
37. 37.What is the smallest number you must add to 684 to make it divisible by 11? (a) 9 (b) 2 (c) 10 (d) 8
38. 38.What is the units digit of 8^27? (a) 4 (b) 8 (c) 6 (d) 2
39. 39.What is the smallest number you must add to 217 to make it divisible by 3? (a) 1 (b) 2
40. 40.What is the smallest number you must add to 358 to make it divisible by 7? (a) 6 (b) 5 (c) 1 (d) 4

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