Question

An architectis desining a house in which a 104-square-foot room and a 130-square-foot room share a wall . what is the greatest possible integer length of the shared wall?

Answer 1

In my experience architects generally avoid number theory.

Let's called the shared wall
`s`. The area is width times height, so the other walls are
`104/s` and
`130/s`. We want the biggest s that gives integer quotients; that would be the greatest common factor of 104 and 130.


`104 = 2^3 13^1`


`140 = 2^1 5^1 13^1`

Greatest common factor and largest shared wall,


`w = 2 * 13 = 26`

That's None of the Above, though 26x4 and 26x5 rooms are pretty goofy, probably why architects don't worry about number theory.