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?

$$ax+by+cz=0$$ 
Let $(a,b,c)=m$
$$a=mA$$
$$b=mB$$
$$c=mC$$
where $(A,B,C)=1$
then
$$Ax+By+Cz=0$$
Then for all integers $u,v,w$, we have:
$$x=Bw-Cv$$
$$y=Cu-Aw$$
$$z=Av-Bu$$
We confirm that
$$A(Bw-Cv)+B(Cu-Aw)+C(Av-Bu)= ABw-ACv+BCu-ABw+ACv-BCu= 0$$
