# Show that if $x^3 + y^3 = z^3$ then 3 divides one of $x,y,z$

I can do this problem by going over all the cases mod 7. Is there any other way to do it?. I think this way is too long , so I wonder if there is another way to do it or some way to simplify my solution.

• A simple modulo argument cannot be "way to long". Nov 25, 2017 at 9:45

According to Fermat's Last Theorem, your equation has no whole number solutions (besides $0$ of course).
• Certainly, but because of this, surely the OP was assuming that $x$, $y$ and $z$ are $3$-adic integers? Nov 25, 2017 at 8:21