Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 766 · 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 369 is divided by 8? (a) 0 (b) 1 (c) 7 (d) 2
2. 2.What is the units digit of 4^17? (a) 6 (b) 3 (c) 5 (d) 4
3. 3.Which of these numbers is divisible by 8? (a) 2335 (b) 2338 (c) 2336 (d) 2348
4. 4.What is the remainder when 164 is divided by 9? (a) 3 (b) 7 (c) 1 (d) 2
5. 5.What is the smallest number you must add to 173 to make it divisible by 4? (a) 1 (b) 2 (c) 3
6. 6.What is the smallest number you must add to 488 to make it divisible by 7? (a) 2 (b) 5 (c) 1 (d) 3
7. 7.Which of these numbers is divisible by 6? (a) 281 (b) 275 (c) 276 (d) 286
8. 8.What is the remainder when 600 is divided by 9? (a) 3 (b) 7 (c) 6 (d) 5
9. 9.What is the remainder when 481 is divided by 6? (a) 5 (b) 2 (c) 1 (d) 0
10. 10.What is the remainder when 548 is divided by 11? (a) 9 (b) 10 (c) 2 (d) 8
11. 11.What is the smallest number you must add to 43 to make it divisible by 4? (a) 1 (b) 2 (c) 3
12. 12.What is the smallest number you must add to 209 to make it divisible by 12? (a) 8 (b) 7 (c) 5 (d) 6
13. 13.What is the remainder when 242 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
14. 14.What is the smallest number you must add to 551 to make it divisible by 11? (a) 8 (b) 9 (c) 10 (d) 1
15. 15.What is the units digit of 9^32? (a) 9 (b) 0 (c) 2 (d) 1
16. 16.What is the smallest number you must add to 80 to make it divisible by 3? (a) 1 (b) 2
17. 17.What is the units digit of 8^45? (a) 6 (b) 2 (c) 4 (d) 8
18. 18.Which of these numbers is divisible by 6? (a) 2504 (b) 2502 (c) 2513 (d) 2499
19. 19.What is the remainder when 270 is divided by 7? (a) 4 (b) 6 (c) 3 (d) 5
20. 20.What is the units digit of 2^48? (a) 4 (b) 6 (c) 2 (d) 8
21. 21.What is the units digit of 7^39? (a) 7 (b) 9 (c) 3 (d) 1
22. 22.What is the smallest number you must add to 380 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
23. 23.What is the units digit of 7^10? (a) 3 (b) 7 (c) 1 (d) 9
24. 24.What is the smallest number you must add to 713 to make it divisible by 10? (a) 3 (b) 6 (c) 8 (d) 7
25. 25.Which of these numbers is divisible by 8? (a) 833 (b) 843 (c) 840 (d) 854
26. 26.What is the units digit of 4^51? (a) 4 (b) 5 (c) 6 (d) 3
27. 27.What is the remainder when 463 is divided by 12? (a) 6 (b) 7 (c) 8 (d) 5
28. 28.What is the smallest number you must add to 163 to make it divisible by 4? (a) 2 (b) 1 (c) 3
29. 29.What is the remainder when 318 is divided by 7? (a) 2 (b) 3 (c) 5 (d) 4
30. 30.Which of these numbers is divisible by 4? (a) 3425 (b) 3424 (c) 3421 (d) 3431
31. 31.What is the remainder when 231 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
32. 32.What is the units digit of 3^10? (a) 7 (b) 9 (c) 1 (d) 3
33. 33.What is the smallest number you must add to 382 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
34. 34.Which of these numbers is divisible by 4? (a) 2476 (b) 2482 (c) 2473 (d) 2478
35. 35.Which of these numbers is divisible by 3? (a) 1426 (b) 1422 (c) 1420 (d) 1423
36. 36.Which of these numbers is divisible by 11? (a) 3151 (b) 3172 (c) 3162 (d) 3157
37. 37.What is the remainder when 377 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
38. 38.What is the smallest number you must add to 611 to make it divisible by 10? (a) 8 (b) 7 (c) 1 (d) 9
39. 39.Which of these numbers is divisible by 4? (a) 3530 (b) 3521 (c) 3526 (d) 3524
40. 40.Which of these numbers is divisible by 3? (a) 1537 (b) 1533 (c) 1532 (d) 1535

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