There do not exist integer $m$ and $n$ such that $12m+15n=1$
It seem obvious but I am not sure how to show this.
If there exist integer m and n such that $12M+15n=1$ then $m$ and $n$ are both positive.
Well clearly the antecent if false on this one. So nothing can really be proved.
But I am not sure how to show this.