Problem

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Tags: modular arithmetic, number theory proposed, number theory



suppose that $a=3^{100}$ and $b=5454$. how many $z$s in $[1,3^{99})$ exist such that for every $c$ that $gcd(c,3)=1$, two equations $x^z\equiv c$ and $x^b\equiv c$ (mod $a$) have the same number of answers?($\frac{100}{6}$ points)