Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 90 · 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 4^45? (a) 4 (b) 5 (c) 3 (d) 6
2. 2.What is the remainder when 145 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
3. 3.What is the smallest number you must add to 337 to make it divisible by 12? (a) 11 (b) 9 (c) 10 (d) 1
4. 4.What is the smallest number you must add to 727 to make it divisible by 12? (a) 4 (b) 6 (c) 5 (d) 7
5. 5.Which of these numbers is divisible by 3? (a) 995 (b) 998 (c) 992 (d) 993
6. 6.Which of these numbers is divisible by 9? (a) 4876 (b) 4857 (c) 4861 (d) 4860
7. 7.What is the smallest number you must add to 194 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
8. 8.What is the smallest number you must add to 94 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 1
9. 9.What is the smallest number you must add to 137 to make it divisible by 4? (a) 3 (b) 1 (c) 2
10. 10.Which of these numbers is divisible by 6? (a) 593 (b) 588 (c) 596 (d) 587
11. 11.What is the units digit of 2^15? (a) 4 (b) 6 (c) 8 (d) 2
12. 12.What is the smallest number you must add to 307 to make it divisible by 10? (a) 7 (b) 3 (c) 4 (d) 2
13. 13.What is the remainder when 507 is divided by 9? (a) 6 (b) 3 (c) 4 (d) 2
14. 14.What is the remainder when 594 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
15. 15.What is the smallest number you must add to 132 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 6
16. 16.What is the units digit of 9^19? (a) 1 (b) 7 (c) 8 (d) 9
17. 17.What is the smallest number you must add to 116 to make it divisible by 10? (a) 5 (b) 4 (c) 6 (d) 3
18. 18.What is the smallest number you must add to 280 to make it divisible by 12? (a) 7 (b) 9 (c) 8 (d) 4
19. 19.What is the remainder when 754 is divided by 12? (a) 2 (b) 9 (c) 10 (d) 11
20. 20.Which of these numbers is divisible by 3? (a) 1154 (b) 1150 (c) 1148 (d) 1149
21. 21.What is the remainder when 400 is divided by 12? (a) 4 (b) 5 (c) 3 (d) 8
22. 22.Which of these numbers is divisible by 6? (a) 2287 (b) 2285 (c) 2286 (d) 2293
23. 23.Which of these numbers is divisible by 9? (a) 2879 (b) 2881 (c) 2893 (d) 2880
24. 24.What is the remainder when 103 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
25. 25.What is the units digit of 3^35? (a) 7 (b) 3 (c) 1 (d) 9
26. 26.Which of these numbers is divisible by 6? (a) 2467 (b) 2477 (c) 2463 (d) 2466
27. 27.Which of these numbers is divisible by 6? (a) 4267 (b) 4256 (c) 4260 (d) 4263
28. 28.Which of these numbers is divisible by 3? (a) 802 (b) 809 (c) 806 (d) 804
29. 29.What is the units digit of 2^36? (a) 2 (b) 4 (c) 8 (d) 6
30. 30.What is the units digit of 8^15? (a) 4 (b) 2 (c) 6 (d) 8
31. 31.What is the units digit of 3^56? (a) 3 (b) 9 (c) 7 (d) 1
32. 32.What is the remainder when 397 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
33. 33.What is the units digit of 7^42? (a) 1 (b) 7 (c) 9 (d) 3
34. 34.Which of these numbers is divisible by 3? (a) 1147 (b) 1144 (c) 1143 (d) 1141
35. 35.What is the smallest number you must add to 97 to make it divisible by 7? (a) 6 (b) 2 (c) 1 (d) 3
36. 36.What is the remainder when 112 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
37. 37.Which of these numbers is divisible by 9? (a) 7702 (b) 7686 (c) 7689 (d) 7680
38. 38.What is the smallest number you must add to 391 to make it divisible by 6? (a) 5 (b) 1 (c) 4 (d) 3
39. 39.What is the remainder when 132 is divided by 10? (a) 1 (b) 8 (c) 2 (d) 3
40. 40.What is the remainder when 214 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3

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