Let $x$ be the number of cookies eaten by Andy.
Since Alexa eats a positive multiple of the number of cookies eaten by Andy, let's say she eats $kx$ cookies, where $k$ is a positive integer.
We know that $x + kx = 24$ (since the siblings finish all 24 cookies).
This simplifies to $(k+1)x = 24$.
Since $x$ and $k+1$ are both positive integers, the maximum value for $x$ occurs when $k+1$ is as small as possible.
The smallest possible value for $k+1$ is 2, which means $x = \frac{24}{2} = 12$.
Therefore, the maximum number of cookies that Andy could have eaten is $\boxed{12}$.The answer is: 12
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Two siblings, Andy and Alexa, bake 24 cookies. Alexa eats some number of cookies that is a positive multiple of the number of cookies eaten by her brother. If the siblings finish all 24 cookies, then what is the maximum number of cookies that the brother, Andy, could have eaten?

### Answer:
