Skip over navigation
This is not written in to the problem itself because EVERY time we try to solve a problem, unless it is very easy, we should think about first trying simple cases.
The problem Em'power'ed also focusses on equating the powers of prime factors.
Factorial Fun
Age 16 to 18
Challenge Level 
Why do this problem?
The problem does not require much knowledge but it calls for careful reasoning and accurate use of standard notation. The problem provides scaffolding steps to help the problem solver think through the ideas needed to solve the problem.Possible approach
Ask the class to count all the factors of $n!$ for $n = 2 , 3, 4$ and $5$ and suggest that they look for the most efficient way to do this. Then suggest that this problem might help them find the best way to do it.This is not written in to the problem itself because EVERY time we try to solve a problem, unless it is very easy, we should think about first trying simple cases.
Key questions
How do we use the fact $24=2^3 \times 3$ to deduce that $24$ has $8$ factors? (Combinatorics question)If we find the prime factors of $n$ how do we find out how many times these prime factors are repeated in $n!$ ?
Possible support
Try the problems Fac-finding, Powerful Factorial and Factoring Factorials. They are all special cases of this problem.The problem Em'power'ed also focusses on equating the powers of prime factors.
You may also like
Old Nuts
In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?
Em'power'ed
Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?