Prove that for any integers $x,y,z$ there exist $a,b,c$ such that $ax+by+cz=0$ It is rather obvious that for any 3 coprime integers $x,y,z$ there exist 3 non-zero integers $a,b,c$ such that:
$$ax+by+cz=0$$
Any simple argument to prove it?

Assume $z\neq 0$. One of them needs to be for $x,y,z$ to be coprime.
Find $u,v\neq 0$ so that $ux+vy\neq 0$. Then let $a=zu,b=zv,c=-(ux+vy)$.
There is always such $u,v$ unless one of $x,y$ is zero. Assume $x=0$ and $y\neq 0$ then choose $a=1,b=z,c=-y$.
If $z=1$ and $x=y=0$, then you can't find a solution with non-zero $c$.
You don't really need coprime, of course. Just that at least $2$ of $x,y,z$ are non-zero. Also, if all three are zero, you can trivially solve with $a=b=c=1$.
