Label the large rectangle by $R$ and place it in the plane so that its lower-left corner is at $(0, 0)$ with sides parallel to $Ox, Oy$.
Let $S$ be the set of vertices of the small rectangles whose both coordinates are integers, and let $T$ be the set of all the small rectangles. We form a bipartite graph on the vertex set $S \cup T$ by joining each point in $S$ to every rectangle in $T$ of which it is a vertex. Observe that each small rectangle has either 0, 2, or 4 vertices in $S$ (because of integer length side), so it contributes an even number of edges. Hence the total number of edges in this bipartite graph is even. On the other hand, each point in $S$ that is not a corner of $R$ is a vertex of exactly 2 or 4 small rectangles, and so also has even degree.
Since $(0, 0)$ is in $S$ and is a corner of exactly one small rectangle, it has odd degree. Therefore there must be another vertex in $S$ of odd degree. The only vertices of odd degree can be corners of $R$, so at least one other corner of $R$ lies in $S$. In particular, either the width or the height of $R$ is an integer. $\Box$