The two numbers

Given two positive integers $a$ and $b$, we define $f(a, b) = \text{lcm}(a, b) - \gcd(a, b)$, where $\text{lcm}$ denotes the [lowest common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) and $\gcd$ denotes the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor). Chef has a positive integer $N$. He wonders, what is the maximum value of $f(a, b)$ over al