how to prove that $2^n = (n+1)(n+2)(n+3)$ has no solution when $n>0$?

Question

I'm having trouble proving that $2^n = (n+1)(n+2)(n+3)$ has no solution when $n>0$. I tried showing there is only one critical point for $(n+1)(n+2)(n+3) - 2^n$. But, I couldn't do it. Can anyone help?

Answer 1

$2^n$ is not a multiple of $3$ and $(n+1)(n+2)(n+3)$ is a multiple of $3$. Thus given equation has no solution.