Two rectangular prisms have the same height, but one is 38 times bigger than the other. They all have integer edge lengths and the diagonals on their faces also have integer lengths. What is the height, the length and the width of the bigger prism, knowing that it’s the smallest possible?
If I understand the question correctly, and coded correctly, then the answer is
height $1155$, width and depth are $1008, 1100$ for the smaller one and $6300, 6688$ for the larger one.
searched for every $h$, list out all the possible sizes and check whether there is one that is $38$ times larger than (which I interpret as multiply by a factor of $38$, not $39$) another.
I searched for all $h \leq 10^5$ to verify that it is indeed the smallest one.