Challenge Level

$13$ is the largest prime factor so $n$ is between $13$ and $17$

$7$ twice so one must come from $7$, one from $14$ so $n$ is at least $14$

$5$ three times so $n$ is at least $15$

$2$ fifteen times

Multiples of $2:$

$2,4$(two)$,6,8$(three)$,10,12,$(two)$,14$ makes eleven factors of $2$

So the number $n$ is $16$ (which itself is $2^4$)

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.