Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 730 · 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 smallest number you must add to 234 to make it divisible by 4? (a) 3 (b) 1 (c) 2
2. 2.What is the remainder when 201 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 0
3. 3.What is the remainder when 518 is divided by 10? (a) 8 (b) 2 (c) 9 (d) 7
4. 4.What is the units digit of 3^60? (a) 1 (b) 9 (c) 7 (d) 3
5. 5.Which of these numbers is divisible by 9? (a) 7614 (b) 7610 (c) 7624 (d) 7617
6. 6.What is the units digit of 3^34? (a) 9 (b) 7 (c) 3 (d) 1
7. 7.What is the smallest number you must add to 359 to make it divisible by 7? (a) 2 (b) 4 (c) 6 (d) 5
8. 8.Which of these numbers is divisible by 3? (a) 432 (b) 437 (c) 430 (d) 433
9. 9.Which of these numbers is divisible by 6? (a) 743 (b) 748 (c) 744 (d) 753
10. 10.What is the units digit of 8^15? (a) 2 (b) 8 (c) 6 (d) 4
11. 11.What is the units digit of 2^23? (a) 6 (b) 4 (c) 8 (d) 2
12. 12.What is the remainder when 408 is divided by 7? (a) 1 (b) 2 (c) 5 (d) 3
13. 13.Which of these numbers is divisible by 6? (a) 3327 (b) 3331 (c) 3340 (d) 3330
14. 14.What is the units digit of 8^11? (a) 8 (b) 6 (c) 4 (d) 2
15. 15.What is the units digit of 3^31? (a) 9 (b) 3 (c) 7 (d) 1
16. 16.Which of these numbers is divisible by 9? (a) 7053 (b) 7058 (c) 7047 (d) 7045
17. 17.What is the units digit of 8^19? (a) 2 (b) 4 (c) 8 (d) 6
18. 18.Which of these numbers is divisible by 3? (a) 1229 (b) 1225 (c) 1222 (d) 1224
19. 19.What is the units digit of 2^53? (a) 4 (b) 6 (c) 8 (d) 2
20. 20.What is the smallest number you must add to 37 to make it divisible by 3? (a) 2 (b) 1
21. 21.Which of these numbers is divisible by 3? (a) 328 (b) 330 (c) 332 (d) 335
22. 22.Which of these numbers is divisible by 6? (a) 5275 (b) 5272 (c) 5274 (d) 5283
23. 23.What is the smallest number you must add to 645 to make it divisible by 9? (a) 4 (b) 3 (c) 2 (d) 6
24. 24.What is the remainder when 71 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
25. 25.What is the remainder when 435 is divided by 7? (a) 6 (b) 0 (c) 2 (d) 1
26. 26.Which of these numbers is divisible by 8? (a) 4793 (b) 4803 (c) 4814 (d) 4800
27. 27.Which of these numbers is divisible by 8? (a) 5733 (b) 5723 (c) 5715 (d) 5720
28. 28.What is the units digit of 8^60? (a) 2 (b) 8 (c) 4 (d) 6
29. 29.Which of these numbers is divisible by 9? (a) 1039 (b) 1049 (c) 1030 (d) 1035
30. 30.What is the remainder when 287 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 1
31. 31.Which of these numbers is divisible by 4? (a) 277 (b) 273 (c) 282 (d) 276
32. 32.Which of these numbers is divisible by 3? (a) 718 (b) 725 (c) 720 (d) 722
33. 33.Which of these numbers is divisible by 8? (a) 3962 (b) 3951 (c) 3957 (d) 3952
34. 34.What is the remainder when 321 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
35. 35.Which of these numbers is divisible by 11? (a) 6006 (b) 6010 (c) 6021 (d) 6002
36. 36.What is the smallest number you must add to 394 to make it divisible by 8? (a) 6 (b) 7 (c) 2 (d) 5
37. 37.What is the smallest number you must add to 46 to make it divisible by 3? (a) 1 (b) 2
38. 38.Which of these numbers is divisible by 11? (a) 6651 (b) 6641 (c) 6644 (d) 6665
39. 39.Which of these numbers is divisible by 4? (a) 447 (b) 451 (c) 448 (d) 453
40. 40.Which of these numbers is divisible by 4? (a) 279 (b) 282 (c) 280 (d) 285

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