Professor Halfbrain has spent his entire weekend with cutting rectangles into smaller rectangles. In particular, he proved the following deep theorem on such dissections.
Professor Halfbrain's rectangle dissection theorem: If a $5\times2$ rectangle is cut into ten smaller rectangles with integer side lengths, then two of these smaller rectangles must be congruent.
This puzzle asks you to improve the professor's theorem and to make it even deeper: Find the largest integer $n$, for which the following statement hold true.
If a $5\times n$ rectangle is cut into ten smaller rectangles with integer side lengths, then two of these smaller rectangles must be congruent.