# Rectangular prism with volume and surface area

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?

• “This is from a timed competition. Fast answers are best”. Would that no be cheating then? Oct 9, 2020 at 19:23
• @JackLeGrüß This is from 2011, why would I cheat
– user807252
Oct 9, 2020 at 19:25
• @JackLeGrüß There are many timed competitions, such as AMC. Have you ever even participated in one?
– user807252
Oct 9, 2020 at 19:25
• @chem1kal: I was enquiring from you,..., not telling you. (No, I have not participated before). Oct 9, 2020 at 19:26

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