Q:

How to prove $19\cdot8^n + 17$ is not a prime number?

Prove $19 \cdot8^n + 17$ is not a prime number. (n is a natural number)

How can I prove this statement?

A:

If $n$ is even, then $19\cdot 8^n+17\equiv 1\cdot (-1)^n-1\equiv 0\pmod 9$.
If $n=4k+1$, then $19\cdot 8^n+17\equiv 9\cdot 1^k+4\equiv 0\pmod{13}$.
If $n\equiv 4k+3$, then $19\cdot 8^n+17\equiv 3\cdot 1^k+2\equiv  0\pmod{5}$.

