<s>[INST] 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.
 [/INST] We have that $\gcd(P(1),P(2))=15$, hence a number $N$ with the property that $N\mid P(n)$ for every $n$ has to be a divisor of $15$. We may notice that $3\mid P(n)$ for every $n$ since exactly one number among $n+3,n+1,n+5$ is a multiple of three. In a similar way, exactly one number among $n+5,n+1,n+7,n+3,n+9$ is a multiple of $5$, hence $\color{red}{15}$ is the answer.
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