2013 AIME II Problems/Problem 7
A group of clerks is assigned the task of sorting files. Each clerk sorts at a constant rate of files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in hours and minutes. Find the number of files sorted during the first one and a half hours of sorting.
There are clerks at the beginning, and clerks are reassigned to another task at the end of each hour. So, , and simplify that we get . Now the problem is to find a reasonable integer solution. Now we know , so divides , AND as long as is a integer, must divide . Now, we suppose that , similarly we get , and so in order to get a minimum integer solution for , it is obvious that works. So we get and . One and a half hour's work should be , so the answer is .
We start with the same approach as solution 1 to get . Then notice that , or , giving the smallest solution at . We find that . Then the number of files they sorted will be
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