I need help with this cuboid volume brain teaser.
A cube with an edge length of 10 is divided into two cuboids with integer edge lengths by a plane cut. Then one of these two cuboids is divided into two cuboids with integer edge lengths by a second plane cut. What is the smallest possible volume of the largest of the three cuboids?
(The answer is $350$.)
So our math teacher found this brainteaser in an old math book of his. We are allowed to work in pairs, and the teacher also gave us the answer to this question.
Now, my friends and I were thinking how this could come to be, and after working with our CAS TInInspire we figured out that the first aspect is to divide the cube in a 7:3 ratio, Otherwise one wouldn't be able to reveive the wanted result. From here we are stuck.