Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 331 · 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 remainder when 108 is divided by 7? (a) 3 (b) 2 (c) 5 (d) 4
2. 2.What is the remainder when 536 is divided by 7? (a) 3 (b) 5 (c) 6 (d) 4
3. 3.What is the units digit of 3^56? (a) 3 (b) 7 (c) 1 (d) 9
4. 4.Which of these numbers is divisible by 3? (a) 2089 (b) 2095 (c) 2092 (d) 2091
5. 5.What is the units digit of 3^48? (a) 7 (b) 9 (c) 3 (d) 1
6. 6.What is the remainder when 201 is divided by 8? (a) 1 (b) 7 (c) 2 (d) 0
7. 7.What is the units digit of 9^18? (a) 2 (b) 0 (c) 1 (d) 9
8. 8.What is the remainder when 125 is divided by 9? (a) 8 (b) 6 (c) 7 (d) 1
9. 9.Which of these numbers is divisible by 6? (a) 4481 (b) 4482 (c) 4493 (d) 4486
10. 10.What is the remainder when 521 is divided by 9? (a) 6 (b) 7 (c) 1 (d) 8
11. 11.What is the smallest number you must add to 378 to make it divisible by 12? (a) 7 (b) 8 (c) 6 (d) 5
12. 12.What is the units digit of 7^50? (a) 3 (b) 7 (c) 1 (d) 9
13. 13.What is the units digit of 2^55? (a) 4 (b) 6 (c) 2 (d) 8
14. 14.What is the remainder when 617 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
15. 15.What is the smallest number you must add to 242 to make it divisible by 3? (a) 2 (b) 1
16. 16.What is the units digit of 2^53? (a) 6 (b) 8 (c) 2 (d) 4
17. 17.What is the smallest number you must add to 239 to make it divisible by 3? (a) 1 (b) 2
18. 18.What is the remainder when 561 is divided by 10? (a) 9 (b) 1 (c) 0 (d) 2
19. 19.What is the units digit of 8^26? (a) 2 (b) 4 (c) 8 (d) 6
20. 20.What is the remainder when 329 is divided by 10? (a) 9 (b) 8 (c) 1 (d) 7
21. 21.What is the remainder when 377 is divided by 9? (a) 7 (b) 8 (c) 1 (d) 6
22. 22.Which of these numbers is divisible by 3? (a) 1193 (b) 1190 (c) 1195 (d) 1191
23. 23.What is the smallest number you must add to 88 to make it divisible by 3? (a) 2 (b) 1
24. 24.What is the remainder when 215 is divided by 12? (a) 1 (b) 10 (c) 9 (d) 11
25. 25.Which of these numbers is divisible by 9? (a) 6976 (b) 6984 (c) 6985 (d) 6998
26. 26.What is the units digit of 9^59? (a) 9 (b) 1 (c) 7 (d) 8
27. 27.Which of these numbers is divisible by 11? (a) 1548 (b) 1534 (c) 1540 (d) 1559
28. 28.What is the remainder when 248 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
29. 29.Which of these numbers is divisible by 9? (a) 626 (b) 634 (c) 621 (d) 619
30. 30.Which of these numbers is divisible by 6? (a) 1541 (b) 1542 (c) 1553 (d) 1546
31. 31.What is the remainder when 364 is divided by 9? (a) 6 (b) 4 (c) 3 (d) 5
32. 32.What is the smallest number you must add to 437 to make it divisible by 8? (a) 3 (b) 4 (c) 2 (d) 5
33. 33.Which of these numbers is divisible by 4? (a) 2731 (b) 2722 (c) 2726 (d) 2724
34. 34.What is the smallest number you must add to 174 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 6
35. 35.What is the remainder when 204 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
36. 36.Which of these numbers is divisible by 11? (a) 9305 (b) 9313 (c) 9291 (d) 9295
37. 37.Which of these numbers is divisible by 6? (a) 2497 (b) 2507 (c) 2513 (d) 2502
38. 38.What is the remainder when 203 is divided by 6? (a) 3 (b) 1 (c) 5 (d) 4
39. 39.What is the units digit of 8^45? (a) 2 (b) 8 (c) 6 (d) 4
40. 40.Which of these numbers is divisible by 11? (a) 1342 (b) 1361 (c) 1346 (d) 1339

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