Rectangular prism with volume and surface area

Question

Here is the question: A rectangular prism has a volume of $720$ cm$^3$ and a surface area of $666$ cm$^2$. If the lengths of all its edges are integers, what is the length of the longest edge?

This is from a previously timed competition. Quick answers will be the most helpful.

You can set up equations and use Vieta's formulas to get $x^3+bx^2+333x-720$. How do I solve the problem after this?

Answer 1

We have $$xyz=720$$ and $$xy+xz+yz=333,$$ where $x$, $y$ and $z$ are naturals.

Now, let $x\geq y\geq z$.

Thus, $$720\geq z^3,$$ which gives $$1\leq z\leq8$$ and $z=3$ is valid, which gives $x=16$ and $y=15.$

Answer 2

Factorize $720$ as a product of three factors, then pairwise, their products sum to $333$. Once you see this, you know that two of the factors must be odd.

From trial and error, I get the longest side to be $16$, with the other two sides being $3$ and $15$. Verifying:

$$3\times15\times16=720,\qquad 2\times(3\times15+15\times16+16\times3)=666.$$