I need to show that every positive integer $n$ can be written uniquely in the form $n = ab$, where $a$ is square-free and $b$ is a square. Then I need to show that $b$ is then the largest square dividing $n$.
The problem I have here is that I can't even see how this is true. How can $1$ be represented this way? Is $1$ a square number? If not, then I cannot see how this is possible. If it is, then I cannot see how $1$, $2$, or $3$ can be represented this way.
Please, any help you can offer giving me a push in the right direction here is greatly appreciated.