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2015 NIMO Summer Contest, 14
Tags: primitive , Residues , Order , number theory , NIMO , summer contest
We say that an integer $a$ is a quadratic, cubic, or quintic residue modulo $n$ if there exists an integer $x$ such that $x^2\equiv a \pmod n$, $x^3 \equiv a \pmod n$, or $x^5 \equiv a \pmod n$, respectively. Further, an integer $a$ is a primitive residue modulo $n$ if it is exactly one of these three types of residues modulo $n$. How many integers $1 \le a \le 2015$ are primitive residues modulo $2015$? [i] Proposed by Michael Ren [/i]