Given surface area of a cuboid, find it's integer side lengths.

Question

Its a rather small problem once you've boiled it down to:

$2(ab + bc + ca) = 100$ (the surface area of said cuboid)

Now, I'm left with the equation:

$ab + bc + ca = 50$,

$3$ variables and $1$ equation. Is there any way to factor or apply a niche formula or similar to solve for lengths: $a, b, c$. Yes, the number is small so you could probably guess and check ( like $1, 2, 16$ works), but is there a proper way to deal with this kind of problem? Thanks.

Answer 1

Let $a\geq b\geq c$.

Thus, $$50\geq3c^2,$$ which gives $$1\leq c\leq 4$$ and we got not so many cases.

For example, for $c=1$ we obtain: $$a+b+ab=50$$ or $$(1+a)(1+b)=51,$$ which gives $$1+a=17$$ and $$1+b=3.$$ The rest is similar.

Answer 2

There is a small finite set of positive solutions. If you allow negative numbers, you get infinitely many solutions, such as $$ (a,b,c) \mapsto (6a+2b+c, 3a+2b, -2a-b) $$