BdMO Online Forum

$350$ Cubic Units

I don't think I need to prove the following statement,

$V \geq \frac{10^3}{3}$[Where V is the required volume] which is quite obvious.
But if you still want to prove it can be easily done with contradiction.

Now the edges of the cuboid must be integers, thus,
$V \geq 334$[Where $V$ is an integer]

Again since the cubes were cut 2 times, so one of the edges of the cuboid must be 10 units making $V$ a multiple of 10,
$\therefore V \geq 340$[where $V$ is a multiple of 10]

Now $340$ can't be a valid cuboid as its factors are $10,2,17$ and all the lengths must be less than or equal to $10$,
$\therefore V \geq 350$

350 can be the volume of a valid cuboid. So we are done! $\square$