Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 816 · 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 210 to make it divisible by 8? (a) 7 (b) 2 (c) 6 (d) 5
2. 2.What is the units digit of 7^23? (a) 7 (b) 1 (c) 3 (d) 9
3. 3.What is the units digit of 9^17? (a) 8 (b) 7 (c) 1 (d) 9
4. 4.What is the units digit of 8^52? (a) 4 (b) 8 (c) 2 (d) 6
5. 5.Which of these numbers is divisible by 3? (a) 656 (b) 652 (c) 654 (d) 659
6. 6.What is the smallest number you must add to 253 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
7. 7.Which of these numbers is divisible by 9? (a) 2984 (b) 2967 (c) 2972 (d) 2970
8. 8.What is the units digit of 9^50? (a) 2 (b) 0 (c) 1 (d) 9
9. 9.Which of these numbers is divisible by 6? (a) 4970 (b) 4979 (c) 4968 (d) 4963
10. 10.What is the remainder when 41 is divided by 3? (a) 1 (b) 0 (c) 2
11. 11.What is the remainder when 383 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
12. 12.What is the smallest number you must add to 191 to make it divisible by 3? (a) 2 (b) 1
13. 13.What is the smallest number you must add to 350 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
14. 14.What is the smallest number you must add to 67 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
15. 15.What is the units digit of 8^49? (a) 2 (b) 8 (c) 6 (d) 4
16. 16.Which of these numbers is divisible by 4? (a) 391 (b) 388 (c) 395 (d) 385
17. 17.Which of these numbers is divisible by 9? (a) 1417 (b) 1428 (c) 1411 (d) 1413
18. 18.Which of these numbers is divisible by 11? (a) 3496 (b) 3501 (c) 3482 (d) 3487
19. 19.What is the units digit of 7^32? (a) 3 (b) 1 (c) 7 (d) 9
20. 20.What is the remainder when 528 is divided by 10? (a) 7 (b) 2 (c) 8 (d) 9
21. 21.What is the smallest number you must add to 286 to make it divisible by 10? (a) 6 (b) 4 (c) 5 (d) 3
22. 22.Which of these numbers is divisible by 3? (a) 478 (b) 474 (c) 475 (d) 473
23. 23.What is the smallest number you must add to 107 to make it divisible by 4? (a) 2 (b) 1 (c) 3
24. 24.What is the units digit of 7^37? (a) 3 (b) 7 (c) 1 (d) 9
25. 25.What is the units digit of 2^48? (a) 4 (b) 8 (c) 6 (d) 2
26. 26.Which of these numbers is divisible by 11? (a) 2015 (b) 2002 (c) 2003 (d) 1996
27. 27.Which of these numbers is divisible by 11? (a) 4041 (b) 4037 (c) 4054 (d) 4036
28. 28.What is the smallest number you must add to 322 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
29. 29.What is the smallest number you must add to 447 to make it divisible by 12? (a) 10 (b) 9 (c) 3 (d) 8
30. 30.What is the smallest number you must add to 218 to make it divisible by 4? (a) 2 (b) 1 (c) 3
31. 31.Which of these numbers is divisible by 9? (a) 7991 (b) 7992 (c) 7997 (d) 8002
32. 32.What is the remainder when 207 is divided by 7? (a) 6 (b) 4 (c) 5 (d) 3
33. 33.Which of these numbers is divisible by 4? (a) 2435 (b) 2426 (c) 2431 (d) 2428
34. 34.What is the units digit of 7^38? (a) 9 (b) 7 (c) 1 (d) 3
35. 35.Which of these numbers is divisible by 3? (a) 725 (b) 719 (c) 722 (d) 720
36. 36.Which of these numbers is divisible by 9? (a) 7506 (b) 7511 (c) 7502 (d) 7522
37. 37.Which of these numbers is divisible by 6? (a) 3482 (b) 3494 (c) 3490 (d) 3486
38. 38.Which of these numbers is divisible by 6? (a) 3595 (b) 3604 (c) 3600 (d) 3608
39. 39.Which of these numbers is divisible by 8? (a) 2094 (b) 2101 (c) 2106 (d) 2096
40. 40.What is the units digit of 8^37? (a) 6 (b) 8 (c) 4 (d) 2

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