<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] As  $P(0)=1\cdot3\cdot5\cdot7\cdot9,$  and $9\nmid P(10),7\nmid P(8)$
The highest divisors of $P(n)$ must divide $3\cdot5$
$P(n)\equiv(n+1)n^2(n+2)^2\pmod3$
Clearly, $3\mid n(n+1)(n+2)$
$P(n)\equiv(n-2)(n-1)n(n+1)(n+2)\pmod5$
Clearly, $5\mid(n-2)(n-1)n(n+1)(n+2)$
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