Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 507 · 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 units digit of 7^26? (a) 7 (b) 9 (c) 3 (d) 1
2. 2.Which of these numbers is divisible by 9? (a) 474 (b) 462 (c) 468 (d) 479
3. 3.What is the units digit of 3^13? (a) 1 (b) 7 (c) 9 (d) 3
4. 4.What is the smallest number you must add to 451 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
5. 5.What is the remainder when 190 is divided by 3? (a) 1 (b) 0 (c) 2
6. 6.What is the remainder when 206 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
7. 7.What is the smallest number you must add to 512 to make it divisible by 9? (a) 2 (b) 8 (c) 1 (d) 3
8. 8.What is the smallest number you must add to 69 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
9. 9.What is the units digit of 3^40? (a) 7 (b) 3 (c) 1 (d) 9
10. 10.What is the units digit of 3^29? (a) 9 (b) 7 (c) 3 (d) 1
11. 11.What is the smallest number you must add to 77 to make it divisible by 3? (a) 2 (b) 1
12. 12.What is the remainder when 69 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
13. 13.Which of these numbers is divisible by 4? (a) 1890 (b) 1886 (c) 1884 (d) 1881
14. 14.What is the units digit of 2^21? (a) 2 (b) 4 (c) 8 (d) 6
15. 15.What is the smallest number you must add to 395 to make it divisible by 6? (a) 2 (b) 3 (c) 1 (d) 5
16. 16.What is the remainder when 162 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
17. 17.What is the smallest number you must add to 543 to make it divisible by 9? (a) 6 (b) 5 (c) 7 (d) 3
18. 18.What is the units digit of 9^16? (a) 1 (b) 9 (c) 0 (d) 2
19. 19.What is the smallest number you must add to 294 to make it divisible by 12? (a) 5 (b) 8 (c) 7 (d) 6
20. 20.What is the smallest number you must add to 215 to make it divisible by 3? (a) 2 (b) 1
21. 21.What is the remainder when 578 is divided by 12? (a) 2 (b) 3 (c) 10 (d) 1
22. 22.Which of these numbers is divisible by 6? (a) 2810 (b) 2805 (c) 2800 (d) 2802
23. 23.What is the smallest number you must add to 203 to make it divisible by 3? (a) 1 (b) 2
24. 24.Which of these numbers is divisible by 11? (a) 8987 (b) 8992 (c) 9005 (d) 8984
25. 25.What is the remainder when 169 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
26. 26.What is the smallest number you must add to 286 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
27. 27.What is the remainder when 484 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
28. 28.What is the remainder when 513 is divided by 8? (a) 7 (b) 0 (c) 2 (d) 1
29. 29.Which of these numbers is divisible by 6? (a) 2737 (b) 2736 (c) 2735 (d) 2743
30. 30.Which of these numbers is divisible by 11? (a) 7679 (b) 7667 (c) 7657 (d) 7674
31. 31.Which of these numbers is divisible by 9? (a) 3909 (b) 3919 (c) 3905 (d) 3906
32. 32.What is the smallest number you must add to 130 to make it divisible by 8? (a) 2 (b) 7 (c) 5 (d) 6
33. 33.What is the remainder when 304 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
34. 34.What is the units digit of 3^53? (a) 7 (b) 9 (c) 3 (d) 1
35. 35.What is the smallest number you must add to 493 to make it divisible by 12? (a) 10 (b) 1 (c) 11 (d) 9
36. 36.Which of these numbers is divisible by 11? (a) 2618 (b) 2621 (c) 2632 (d) 2617
37. 37.What is the remainder when 167 is divided by 10? (a) 7 (b) 6 (c) 8 (d) 3
38. 38.Which of these numbers is divisible by 6? (a) 4316 (b) 4311 (c) 4303 (d) 4308
39. 39.Which of these numbers is divisible by 9? (a) 5763 (b) 5747 (c) 5757 (d) 5751
40. 40.What is the smallest number you must add to 116 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4

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