Common power residues from coprime exponent congruences ($n\equiv 3x^3\equiv 5y^5\equiv 7z^7(\mathrm{mod}p),p\equiv 1(\mathrm{mod}105)$)

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.