Largest divisor of $P(n) = (n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$ 
Let $P(n) = (n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$. What is the largest integer that is a divisor of $P(n)$ for all positive even integers $n$?

I can see that obviously it is divisible by $3$ and $5$ and hence answer is $15$ but I can't prove it.

In order to prove that $5$ is a divisor of $P(n)$ for even $n$, consider the following cases:


*

*$n\equiv\color\red0\pmod{10} \implies n+5\equiv\color\red0+5\equiv5\pmod{10} \implies 5|P(n)$

*$n\equiv\color\red2\pmod{10} \implies n+3\equiv\color\red2+3\equiv5\pmod{10} \implies 5|P(n)$

*$n\equiv\color\red4\pmod{10} \implies n+1\equiv\color\red4+1\equiv5\pmod{10} \implies 5|P(n)$

*$n\equiv\color\red6\pmod{10} \implies n+9\equiv\color\red6+9\equiv15\equiv5\pmod{10} \implies 5|P(n)$

*$n\equiv\color\red8\pmod{10} \implies n+7\equiv\color\red8+7\equiv15\equiv5\pmod{10} \implies 5|P(n)$


That said, I'm not sure how to prove that $5$ is indeed the largest divisor of $P(n)$...
