ELMO 2023 Problem 2

Let $a$, $b$, and $n$ be positive integers. A lemonade stand owns $n$ cups, all of which are initially empty. The lemonade stand has a filling machine and an emptying machine, which operate according to the following rules:

• If at any moment, $a$ completely empty cups are available, the filling machine spends the next $a$ minutes filling those $a$ cups simultaneously and doing nothing else.
• If at any moment, $b$ completely full cups are available, the emptying machine spends the next $b$ minutes emptying those $b$ cups simultaneously and doing nothing else. Suppose that after a sufficiently long time has passed, both the filling machine and emptying machine work without pausing. Find, in terms of $a$ and $b$, the least possible value of $n$.

Proposed by Raymond Feng