This shouldn't be difficult but I can't get it out..:
Let $c,d,N$ be integers with $\operatorname{hcf}(c,d,N) = 1$. Show that there exist $m, n ∈\mathbb Z$ with $\operatorname{hcf}(c+mN,d+nN) = 1$
Any help appreciated!
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For all $,n\in \Bbb Z!:,$ $(c,N,d+nN) = (c,N,d)=1,,$ so $,(c+mN,d+nN)=1$ for some $,m\in \Bbb Z$ by the dupe. $\ \ $ – Bill Dubuque Sep 15 '23 at 19:13