2016 AMC 12B Problems/Problem 22
Problem
For a certain positive integer less than , the decimal equivalent of is , a repeating decimal of period of , and the decimal equivalent of is , a repeating decimal of period . In which interval does lie?
Solution
Solution by e_power_pi_times_i
If , must be a factor of . Also, by the same procedure, must be a factor of . Checking through all the factors of and that are less than , we see that is a solution, so the answer is .
Note: and are both solutions, which invalidates this method. However, we need to examine all factors of that are not factors of , , or , or . Additionally, we need to be a factor of but not , , or . Indeed, satisfies these requirements.
We can see that and are not solutions by checking it in the requirements of the problem: , period 1, and , period 3. Thus, is the only answer.
For anyone who wants more information about repeating decimals, visit: https://en.wikipedia.org/wiki/Repeating_decimal
See Also
2016 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.