integers - Use theory of congruence to prove.....

Use the theory of congruence to prove that $17|(2^{3n+1} +3\times5^{2n+1})$ for all integer $n\geq1$
$(2^{3n+1} +3\times5^{2n+1})$=$2\times8^n+15\times25^n$
=$17\times8^{n-1}+374\times25^{n-1}+25^{n-1}-8^{n-1}$

=$25^{n-1}-8^{n-1}$
=$8^{n-1}-8^{n-1}$ [since $25\equiv 8\mod 17)$
=$0\mod 17$

$0$ is divisible by $17$

Is this correct?