I'm stuck on a homework problem and I'm hoping to get a hint in the right direction.
Assuming that $a\perp b$ ($a$ coprime with $b$), I would like to show that for all integers $n$, there is a nonzero integer $k$ so that $a+bk\perp n$.
I started by writing the ideal $(a+bk,n)=\{ax+bkx+ny : x,y\in\mathbb{Z}\}$, hoping to find a $k$ so that $(a+bk,n)=(1)=\mathbb{Z}$. If $a\perp n$, then setting $k=0$ works. However, I had no luck finding such a $k$ when $a\not\perp n$. (This doesn't work. See Brian's comment.)