Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 369 · 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.Which of these numbers is divisible by 3? (a) 2087 (b) 2085 (c) 2089 (d) 2084
2. 2.Which of these numbers is divisible by 3? (a) 1462 (b) 1465 (c) 1464 (d) 1469
3. 3.What is the smallest number you must add to 173 to make it divisible by 4? (a) 2 (b) 1 (c) 3
4. 4.What is the units digit of 7^29? (a) 9 (b) 7 (c) 1 (d) 3
5. 5.Which of these numbers is divisible by 8? (a) 1989 (b) 2006 (c) 1999 (d) 1992
6. 6.What is the units digit of 2^47? (a) 8 (b) 4 (c) 2 (d) 6
7. 7.What is the smallest number you must add to 230 to make it divisible by 3? (a) 1 (b) 2
8. 8.What is the units digit of 9^33? (a) 8 (b) 1 (c) 7 (d) 9
9. 9.Which of these numbers is divisible by 3? (a) 1991 (b) 1984 (c) 1987 (d) 1986
10. 10.Which of these numbers is divisible by 11? (a) 6545 (b) 6549 (c) 6540 (d) 6566
11. 11.What is the remainder when 308 is divided by 9? (a) 1 (b) 2 (c) 7 (d) 3
12. 12.What is the units digit of 4^15? (a) 6 (b) 3 (c) 5 (d) 4
13. 13.What is the remainder when 310 is divided by 9? (a) 5 (b) 4 (c) 3 (d) 6
14. 14.Which of these numbers is divisible by 6? (a) 1442 (b) 1434 (c) 1435 (d) 1431
15. 15.What is the remainder when 270 is divided by 8? (a) 7 (b) 6 (c) 5 (d) 2
16. 16.Which of these numbers is divisible by 8? (a) 584 (b) 588 (c) 580 (d) 598
17. 17.Which of these numbers is divisible by 3? (a) 2047 (b) 2049 (c) 2050 (d) 2053
18. 18.What is the units digit of 2^26? (a) 6 (b) 2 (c) 4 (d) 8
19. 19.Which of these numbers is divisible by 4? (a) 2379 (b) 2385 (c) 2381 (d) 2380
20. 20.What is the units digit of 4^23? (a) 5 (b) 4 (c) 6 (d) 3
21. 21.Which of these numbers is divisible by 6? (a) 497 (b) 507 (c) 499 (d) 498
22. 22.What is the units digit of 3^38? (a) 3 (b) 7 (c) 1 (d) 9
23. 23.What is the units digit of 8^51? (a) 2 (b) 4 (c) 6 (d) 8
24. 24.What is the smallest number you must add to 367 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
25. 25.Which of these numbers is divisible by 11? (a) 4925 (b) 4909 (c) 4930 (d) 4917
26. 26.What is the units digit of 9^21? (a) 8 (b) 9 (c) 1 (d) 7
27. 27.What is the remainder when 255 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
28. 28.What is the remainder when 433 is divided by 12? (a) 1 (b) 11 (c) 2 (d) 0
29. 29.What is the smallest number you must add to 361 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
30. 30.What is the remainder when 776 is divided by 12? (a) 7 (b) 4 (c) 9 (d) 8
31. 31.What is the remainder when 201 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 2
32. 32.Which of these numbers is divisible by 9? (a) 4935 (b) 4951 (c) 4948 (d) 4941
33. 33.What is the smallest number you must add to 149 to make it divisible by 3? (a) 1 (b) 2
34. 34.Which of these numbers is divisible by 11? (a) 2681 (b) 2673 (c) 2666 (d) 2687
35. 35.Which of these numbers is divisible by 8? (a) 3881 (b) 3873 (c) 3869 (d) 3872
36. 36.What is the smallest number you must add to 169 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 1
37. 37.What is the remainder when 552 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
38. 38.Which of these numbers is divisible by 3? (a) 1327 (b) 1330 (c) 1326 (d) 1325
39. 39.What is the smallest number you must add to 131 to make it divisible by 3? (a) 2 (b) 1
40. 40.Which of these numbers is divisible by 3? (a) 1679 (b) 1685 (c) 1680 (d) 1681

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