Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 226 · 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 376 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
2. 2.What is the units digit of 9^14? (a) 0 (b) 2 (c) 1 (d) 9
3. 3.What is the remainder when 578 is divided by 9? (a) 7 (b) 3 (c) 2 (d) 1
4. 4.What is the units digit of 3^27? (a) 1 (b) 7 (c) 3 (d) 9
5. 5.Which of these numbers is divisible by 3? (a) 2599 (b) 2596 (c) 2593 (d) 2595
6. 6.What is the smallest number you must add to 601 to make it divisible by 12? (a) 1 (b) 11 (c) 9 (d) 10
7. 7.Which of these numbers is divisible by 3? (a) 1771 (b) 1769 (c) 1767 (d) 1766
8. 8.What is the smallest number you must add to 178 to make it divisible by 3? (a) 2 (b) 1
9. 9.What is the units digit of 9^49? (a) 1 (b) 7 (c) 8 (d) 9
10. 10.What is the smallest number you must add to 164 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
11. 11.What is the units digit of 7^27? (a) 7 (b) 1 (c) 3 (d) 9
12. 12.What is the units digit of 3^52? (a) 1 (b) 9 (c) 7 (d) 3
13. 13.What is the smallest number you must add to 274 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
14. 14.Which of these numbers is divisible by 11? (a) 8844 (b) 8847 (c) 8860 (d) 8837
15. 15.What is the units digit of 2^57? (a) 2 (b) 6 (c) 8 (d) 4
16. 16.Which of these numbers is divisible by 8? (a) 766 (b) 760 (c) 773 (d) 759
17. 17.Which of these numbers is divisible by 4? (a) 3041 (b) 3050 (c) 3046 (d) 3044
18. 18.What is the units digit of 9^31? (a) 9 (b) 7 (c) 8 (d) 1
19. 19.What is the remainder when 423 is divided by 12? (a) 9 (b) 3 (c) 2 (d) 4
20. 20.What is the smallest number you must add to 128 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
21. 21.What is the units digit of 8^48? (a) 4 (b) 6 (c) 8 (d) 2
22. 22.What is the smallest number you must add to 379 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 1
23. 23.Which of these numbers is divisible by 4? (a) 652 (b) 654 (c) 657 (d) 651
24. 24.What is the units digit of 3^21? (a) 3 (b) 7 (c) 9 (d) 1
25. 25.What is the units digit of 4^54? (a) 4 (b) 6 (c) 7 (d) 5
26. 26.What is the units digit of 2^22? (a) 2 (b) 8 (c) 4 (d) 6
27. 27.What is the remainder when 778 is divided by 12? (a) 11 (b) 9 (c) 10 (d) 2
28. 28.What is the remainder when 426 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 1
29. 29.What is the units digit of 7^51? (a) 1 (b) 7 (c) 9 (d) 3
30. 30.Which of these numbers is divisible by 8? (a) 1928 (b) 1929 (c) 1922 (d) 1940
31. 31.Which of these numbers is divisible by 4? (a) 2919 (b) 2915 (c) 2916 (d) 2921
32. 32.Which of these numbers is divisible by 11? (a) 5174 (b) 5183 (c) 5181 (d) 5198
33. 33.What is the remainder when 477 is divided by 11? (a) 3 (b) 4 (c) 7 (d) 5
34. 34.What is the remainder when 497 is divided by 12? (a) 4 (b) 7 (c) 6 (d) 5
35. 35.What is the smallest number you must add to 593 to make it divisible by 11? (a) 10 (b) 1 (c) 3 (d) 2
36. 36.What is the units digit of 2^25? (a) 8 (b) 6 (c) 2 (d) 4
37. 37.Which of these numbers is divisible by 4? (a) 1099 (b) 1102 (c) 1093 (d) 1096
38. 38.Which of these numbers is divisible by 8? (a) 3022 (b) 3010 (c) 3006 (d) 3008
39. 39.What is the smallest number you must add to 545 to make it divisible by 12? (a) 7 (b) 5 (c) 8 (d) 6
40. 40.Which of these numbers is divisible by 4? (a) 1005 (b) 1002 (c) 1000 (d) 997

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