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Factorised Factorial

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2
$n!$ is divisible by $5^3$ so $n!$ must be at least $15$. But $15!$ is only divisible by $2^{11}$ so $n$ is not $15$. $n!$ is not divisible by $17$ so $n$ is less than $17$. Hence $n=16$.

This problem is taken from the UKMT Mathematical Challenges.
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