Fifth power remainders

Number Theory Level 5

For any prime \(p>7,\) consider the remainders when the fifth powers (i.e. \( n^5\)) are divided by \(p\). Let \(f(p)\) be the smallest possible remainder which is not 0 or 1.

The sum of the two smallest possible ratios \(\frac{p}{f(p)}\) can be written as \( \frac{a}{b} \), where \(a\) and \(b\) are coprime positive integers. What are the last three digits of the value of \(a+b\)?

Details and assumptions

As an explicit example, for \(p=13\), you can verify that the possible remainders are \(0, 1, 2, \ldots, 12 \). Thus \( f(13) = 2 \).


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