Show that $3^n-2^n\cdot 5$ is composite for infinitely many $n$

Question

I came across this problem:

Show that $3^n-2^n\cdot 5$ is composite for infinitely many $n$

and do not know how to solve it. I only know that it is true for $n=7$, since then $1547=17\cdot 91$.

Answer 1

Actually you already did most of the work. You know that $3^n-2^n\cdot5$ is divisible by $17$ for $n=7$.

Now, consider $n=16+7$. What does Fermat's Little Theorem tell us?