Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 376 · 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 4? (a) 2004 (b) 2009 (c) 2007 (d) 2003
2. 2.What is the units digit of 4^28? (a) 4 (b) 6 (c) 7 (d) 5
3. 3.What is the remainder when 44 is divided by 3? (a) 0 (b) 1 (c) 2
4. 4.What is the smallest number you must add to 472 to make it divisible by 6? (a) 1 (b) 4 (c) 2 (d) 3
5. 5.What is the remainder when 246 is divided by 9? (a) 6 (b) 2 (c) 3 (d) 4
6. 6.What is the units digit of 3^28? (a) 3 (b) 7 (c) 1 (d) 9
7. 7.Which of these numbers is divisible by 8? (a) 3349 (b) 3359 (c) 3344 (d) 3343
8. 8.Which of these numbers is divisible by 8? (a) 4472 (b) 4467 (c) 4487 (d) 4476
9. 9.What is the smallest number you must add to 113 to make it divisible by 4? (a) 1 (b) 2 (c) 3
10. 10.What is the units digit of 9^36? (a) 0 (b) 9 (c) 2 (d) 1
11. 11.Which of these numbers is divisible by 6? (a) 756 (b) 759 (c) 751 (d) 766
12. 12.Which of these numbers is divisible by 6? (a) 4912 (b) 4903 (c) 4918 (d) 4908
13. 13.What is the units digit of 9^19? (a) 1 (b) 8 (c) 9 (d) 7
14. 14.Which of these numbers is divisible by 3? (a) 580 (b) 584 (c) 582 (d) 587
15. 15.Which of these numbers is divisible by 11? (a) 3311 (b) 3323 (c) 3309 (d) 3321
16. 16.What is the remainder when 310 is divided by 9? (a) 4 (b) 3 (c) 5 (d) 6
17. 17.What is the units digit of 7^40? (a) 9 (b) 7 (c) 1 (d) 3
18. 18.What is the units digit of 9^31? (a) 8 (b) 7 (c) 1 (d) 9
19. 19.What is the units digit of 8^30? (a) 6 (b) 2 (c) 8 (d) 4
20. 20.Which of these numbers is divisible by 11? (a) 9754 (b) 9757 (c) 9764 (d) 9773
21. 21.Which of these numbers is divisible by 6? (a) 3420 (b) 3419 (c) 3424 (d) 3428
22. 22.What is the remainder when 438 is divided by 9? (a) 6 (b) 7 (c) 5 (d) 3
23. 23.What is the units digit of 3^38? (a) 7 (b) 9 (c) 3 (d) 1
24. 24.What is the smallest number you must add to 124 to make it divisible by 3? (a) 2 (b) 1
25. 25.What is the units digit of 9^49? (a) 9 (b) 8 (c) 7 (d) 1
26. 26.Which of these numbers is divisible by 11? (a) 1484 (b) 1503 (c) 1485 (d) 1489
27. 27.What is the units digit of 7^19? (a) 1 (b) 7 (c) 9 (d) 3
28. 28.Which of these numbers is divisible by 6? (a) 3425 (b) 3410 (c) 3416 (d) 3414
29. 29.Which of these numbers is divisible by 8? (a) 4792 (b) 4793 (c) 4806 (d) 4786
30. 30.What is the remainder when 347 is divided by 10? (a) 6 (b) 7 (c) 8 (d) 3
31. 31.What is the units digit of 8^56? (a) 6 (b) 4 (c) 8 (d) 2
32. 32.What is the remainder when 64 is divided by 6? (a) 2 (b) 5 (c) 3 (d) 4
33. 33.Which of these numbers is divisible by 9? (a) 6291 (b) 6284 (c) 6303 (d) 6295
34. 34.What is the units digit of 8^57? (a) 4 (b) 6 (c) 2 (d) 8
35. 35.What is the units digit of 8^16? (a) 4 (b) 6 (c) 2 (d) 8
36. 36.Which of these numbers is divisible by 9? (a) 4505 (b) 4511 (c) 4492 (d) 4500
37. 37.Which of these numbers is divisible by 9? (a) 1446 (b) 1451 (c) 1440 (d) 1439
38. 38.Which of these numbers is divisible by 6? (a) 3567 (b) 3557 (c) 3560 (d) 3558
39. 39.What is the smallest number you must add to 160 to make it divisible by 9? (a) 3 (b) 1 (c) 7 (d) 2
40. 40.Which of these numbers is divisible by 3? (a) 907 (b) 905 (c) 906 (d) 911

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