The following theorem has many proofs, several of which are highlighted in this document.
Whenever a large rectangle is tiled by rectangles, each of which has at least one integer side - the large rectangle has at least one integer side, too.
The document has a proof using prime numbers and scaling:
Prime numbers (Raphael Robinson, Univ. of California, Berkeley) We claim that for each prime $p$, either the height or width of $R$ is within $1/p$ of an integer. It follows that one of these is an integer. To prove the claim, scale the entire tiling up by a factor of $p$ in each direction, and consider the tiling obtained by replacing all tile-corners $(x, y)$ in the scaled-up tiling by $([x], [y])$. This yields an integer-sided rectangle tiled by integer-sided rectangles, each of which has one side a multiple of $p$. Therefore, the area of the large integer-sided rectangle is a multiple of $p$, whence one of its sides must be a multiple of $p$. Moreover, the dimensions of this rectangle differ from the dimensions of the scaled-up rectangle by less than $1$. It follows that R has a side that differs from an integer by less than $1/p$ .
I couldn't figure out why it is necessary to have the condition that $p$ be prime.
