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?
Starting from here: $$\begin{align}2\times8^n+15\times25^n&\equiv2\times8^n+15\times (17+8)^n\equiv\\ &\equiv2\times 8^n+15\times 8^n\equiv\\ &\equiv17\times 8^n\equiv 0 \pmod{17}.\end{align}$$