= Common power residues from coprime exponent congruences
{title2=$n\equiv3x^3\equiv5y^5\equiv7z^7\pmod p,\quad p\equiv1\pmod{105}$}
Choose a <primitive root> $g$ modulo $p$ and write the nonzero coefficients $3,5,7$ as $g^a,g^b,g^c$. The <Chinese remainder theorem> gives an exponent $r$ congruent to $a,b,c$ modulo $3,5,7$, respectively. Then $n=g^r$ and the three roots $g^{(r-a)/3}$, $g^{(r-b)/5}$, $g^{(r-c)/7}$ give the displayed common nonzero residue. This technique converts multiplicative power constraints into simultaneous additive <modular congruences> for exponents.
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