2008 Mock ARML 2 Problems/Problem 7
Let equal the number of zeroes to the right of the rightmost non-zero digit in the decimal form of , and let . Given that can be written as , where and are relatively prime positive integers, is less than , and is less than , find .
Note that is an integer. From Legendre's Formula, we see that
Now note that the largest multiple of 15 that is less than is . Therefore
We do the same process again: The largest multiple of 15 less than is , so
Similarly, . We then see a pattern;
Now note that , so
This shows that , so .
|2008 Mock ARML 2 (Problems, Source)|
|1 • 2 • 3 • 4 • 5 • 6 • 7 • 8|