Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 392 · 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 2^49? (a) 2 (b) 4 (c) 6 (d) 8
2. 2.Which of these numbers is divisible by 3? (a) 1625 (b) 1627 (c) 1626 (d) 1630
3. 3.What is the remainder when 238 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
4. 4.Which of these numbers is divisible by 9? (a) 986 (b) 979 (c) 971 (d) 972
5. 5.What is the units digit of 3^57? (a) 1 (b) 3 (c) 9 (d) 7
6. 6.Which of these numbers is divisible by 4? (a) 1346 (b) 1342 (c) 1340 (d) 1338
7. 7.Which of these numbers is divisible by 8? (a) 5338 (b) 5327 (c) 5328 (d) 5332
8. 8.What is the remainder when 284 is divided by 7? (a) 3 (b) 4 (c) 6 (d) 5
9. 9.What is the smallest number you must add to 242 to make it divisible by 3? (a) 1 (b) 2
10. 10.What is the remainder when 187 is divided by 12? (a) 8 (b) 6 (c) 5 (d) 7
11. 11.What is the units digit of 2^32? (a) 2 (b) 4 (c) 6 (d) 8
12. 12.What is the smallest number you must add to 158 to make it divisible by 3? (a) 2 (b) 1
13. 13.What is the remainder when 382 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 2
14. 14.What is the remainder when 481 is divided by 8? (a) 7 (b) 0 (c) 1 (d) 2
15. 15.Which of these numbers is divisible by 4? (a) 2289 (b) 2283 (c) 2284 (d) 2286
16. 16.What is the smallest number you must add to 139 to make it divisible by 3? (a) 2 (b) 1
17. 17.Which of these numbers is divisible by 6? (a) 3953 (b) 3942 (c) 3937 (d) 3947
18. 18.What is the remainder when 490 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 8
19. 19.What is the units digit of 7^18? (a) 3 (b) 9 (c) 1 (d) 7
20. 20.What is the smallest number you must add to 850 to make it divisible by 11? (a) 7 (b) 9 (c) 8 (d) 3
21. 21.What is the remainder when 151 is divided by 3? (a) 1 (b) 2 (c) 0
22. 22.What is the remainder when 243 is divided by 12? (a) 9 (b) 2 (c) 3 (d) 4
23. 23.Which of these numbers is divisible by 6? (a) 2492 (b) 2487 (c) 2490 (d) 2499
24. 24.Which of these numbers is divisible by 6? (a) 4773 (b) 4779 (c) 4786 (d) 4776
25. 25.Which of these numbers is divisible by 6? (a) 4456 (b) 4452 (c) 4450 (d) 4459
26. 26.What is the units digit of 7^43? (a) 7 (b) 3 (c) 9 (d) 1
27. 27.Which of these numbers is divisible by 3? (a) 1645 (b) 1641 (c) 1642 (d) 1640
28. 28.Which of these numbers is divisible by 4? (a) 2583 (b) 2573 (c) 2576 (d) 2578
29. 29.Which of these numbers is divisible by 8? (a) 394 (b) 378 (c) 384 (d) 387
30. 30.What is the remainder when 436 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
31. 31.What is the units digit of 8^15? (a) 8 (b) 4 (c) 6 (d) 2
32. 32.What is the remainder when 212 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
33. 33.Which of these numbers is divisible by 4? (a) 2877 (b) 2871 (c) 2874 (d) 2872
34. 34.What is the units digit of 8^47? (a) 8 (b) 4 (c) 2 (d) 6
35. 35.What is the smallest number you must add to 326 to make it divisible by 9? (a) 2 (b) 6 (c) 8 (d) 7
36. 36.What is the remainder when 708 is divided by 11? (a) 7 (b) 5 (c) 3 (d) 4
37. 37.What is the units digit of 4^14? (a) 4 (b) 6 (c) 5 (d) 7
38. 38.Which of these numbers is divisible by 8? (a) 1195 (b) 1202 (c) 1212 (d) 1200
39. 39.What is the remainder when 263 is divided by 11? (a) 9 (b) 1 (c) 10 (d) 8
40. 40.What is the units digit of 4^17? (a) 3 (b) 5 (c) 6 (d) 4

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