Cuboid with Integer Edges and Face Diagonals

Theorem
The smallest cuboid whose edges and the diagonals of whose faces are all integers has edge lengths $44$, $117$ and $240$.

Its space diagonal, however, is not an integer.

Proof
The edges are given as having lengths $44$, $117$ and $240$.

The faces are therefore:
 * $44 \times 117$
 * $44 \times 240$
 * $117 \times 240$

The diagonals of these faces are given by Pythagoras's Theorem as follows:

However, its space diagonal is calculated as:

which, as can be seen, is not an integer.