Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 429 · 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 8^35? (a) 4 (b) 8 (c) 6 (d) 2
2. 2.Which of these numbers is divisible by 4? (a) 1049 (b) 1041 (c) 1047 (d) 1044
3. 3.What is the remainder when 243 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
4. 4.What is the remainder when 316 is divided by 5? (a) 4 (b) 0 (c) 2 (d) 1
5. 5.What is the remainder when 97 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
6. 6.Which of these numbers is divisible by 9? (a) 4147 (b) 4154 (c) 4140 (d) 4139
7. 7.What is the smallest number you must add to 74 to make it divisible by 4? (a) 1 (b) 3 (c) 2
8. 8.Which of these numbers is divisible by 9? (a) 7294 (b) 7281 (c) 7274 (d) 7283
9. 9.What is the remainder when 132 is divided by 7? (a) 1 (b) 5 (c) 6 (d) 4
10. 10.Which of these numbers is divisible by 4? (a) 2063 (b) 2059 (c) 2056 (d) 2053
11. 11.What is the remainder when 440 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 2
12. 12.What is the units digit of 4^50? (a) 4 (b) 7 (c) 5 (d) 6
13. 13.Which of these numbers is divisible by 4? (a) 2535 (b) 2536 (c) 2539 (d) 2541
14. 14.What is the remainder when 200 is divided by 9? (a) 3 (b) 7 (c) 1 (d) 2
15. 15.What is the units digit of 3^25? (a) 3 (b) 9 (c) 1 (d) 7
16. 16.What is the units digit of 4^35? (a) 5 (b) 4 (c) 3 (d) 6
17. 17.What is the units digit of 4^38? (a) 7 (b) 5 (c) 6 (d) 4
18. 18.What is the remainder when 503 is divided by 10? (a) 7 (b) 2 (c) 4 (d) 3
19. 19.What is the smallest number you must add to 215 to make it divisible by 10? (a) 5 (b) 7 (c) 6 (d) 4
20. 20.What is the remainder when 269 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
21. 21.What is the smallest number you must add to 673 to make it divisible by 12? (a) 9 (b) 10 (c) 11 (d) 1
22. 22.Which of these numbers is divisible by 4? (a) 222 (b) 219 (c) 216 (d) 214
23. 23.What is the units digit of 2^11? (a) 8 (b) 6 (c) 2 (d) 4
24. 24.What is the units digit of 3^18? (a) 3 (b) 9 (c) 7 (d) 1
25. 25.Which of these numbers is divisible by 9? (a) 1081 (b) 1090 (c) 1101 (d) 1089
26. 26.What is the remainder when 745 is divided by 10? (a) 6 (b) 4 (c) 7 (d) 5
27. 27.What is the units digit of 9^43? (a) 1 (b) 8 (c) 9 (d) 7
28. 28.What is the remainder when 103 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
29. 29.What is the remainder when 237 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 5
30. 30.Which of these numbers is divisible by 8? (a) 6500 (b) 6489 (c) 6488 (d) 6484
31. 31.What is the remainder when 233 is divided by 10? (a) 2 (b) 4 (c) 3 (d) 7
32. 32.What is the remainder when 682 is divided by 10? (a) 1 (b) 2 (c) 3 (d) 8
33. 33.What is the units digit of 9^28? (a) 0 (b) 1 (c) 2 (d) 9
34. 34.What is the remainder when 225 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
35. 35.What is the units digit of 3^28? (a) 1 (b) 9 (c) 3 (d) 7
36. 36.Which of these numbers is divisible by 3? (a) 1691 (b) 1686 (c) 1684 (d) 1687
37. 37.What is the units digit of 8^11? (a) 8 (b) 6 (c) 2 (d) 4
38. 38.What is the smallest number you must add to 629 to make it divisible by 12? (a) 8 (b) 5 (c) 6 (d) 7
39. 39.What is the remainder when 446 is divided by 7? (a) 5 (b) 2 (c) 4 (d) 6
40. 40.What is the remainder when 253 is divided by 7? (a) 1 (b) 0 (c) 2 (d) 6

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