\section*{Problem 3 - 031033}

Two students are given the task to multiply two numbers \(a\) and \(b\) \((a>0, b>0)\). To check their work, they divide the product by the smaller factor. The first student gets 575 with a remainder of 227. The second student gets 572 with a remainder of 308. Each had forgotten to add a 1 at a different place when adding the partial products. Therefore, the first student ended up with 100 less in the result, and the second student ended up with 1000 less.

What are the numbers \(a\) and \(b\)?
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Without loss of generality, let \(b\) be the smaller number. Then, from the given information, the following system of equations arises:

\[
575 b + 227 = a b - 100 \quad ; \quad 572 b + 308 = a b - 1000
\]

This system of equations can be solved, for example, by solving the first equation for \(b\). One obtains \(b = -\frac{327}{575 - a}\).

Substituting this into the second equation leads, after equivalent transformations, to the solution \(a = 576\). From this, using the relationship \(b = -\frac{327}{575 - a} = 327\).