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For a positive integer $n$, we define $n!$ to be the product of all the positive integers from $1$ to $n$; that is $n!=1\times 2\times 3\times\ldots\times n$.

If $n!=2^{15}\times 3^6\times 5^3\times 7^2\times 11\times 13$, what is the value of $n$?

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This problem is taken from the UKMT Mathematical Challenges.