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Factorised Factorial

Age 14 to 16 Short
Challenge Level

Answer: $n=16$

$13$ 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.