$n!$ 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.**View the archive of all weekly problems grouped by curriculum topic*