1 An integer is a perfect number if and only if it is equal to the sum of all of its divisors except itself. For example, 28 is a perfect number since $28=1+2+4+7+14$.
Let $n$ ! denote the product $1 \cdot 2 \cdot 3 \cdots n$, where $n$ is a positive integer. An integer is a factorial if and only if it is equal to $n!$ for some positive integer $n$. For example, 24 is a factorial number since $24=4!=1 \cdot 2 \cdot 3 \cdot 4$. Find all perfect numbers greater than 1 that are also factorials.
Solution: The only perfect factorial is $6=3$ !. Certainly, $2!=2$ is not perfect. For $n>3$, note that $n!=6 k$, where $k>1$, and thus the factors of $n!$ will include $1, k, 2 k, 3 k$. This sums to $6 k+1$, showing that $n$ ! is not perfect.