2016 AMC 10B Problems/Problem 25
Problem
Let , where denotes the greatest integer less than or equal to . How many distinct values does assume for ?
Solution
Since , we have
The function can then be simplified into
which becomes
We can see that for each value of k, can equal integers from 0 to k-1.
Clearly, the value of changes only when x is equal to any of the fractions .
So we want to count how many distinct fractions have the form where . We can find this easily by computing where is the Euler Totient Function. Basically counts the number of fractions with as its denominator (after simplification). This comes out to be .
Because the value of is at least 0 and can increase 31 times, there are a total of different possible values of .
See Also
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