Let $m$ be a positive integer. Let $a,b$ be integers with $0 \leq a,b < m$, $a,b$ not both zero, $\gcd(a,b,m)=1$.
Do there necessarily exist integers $x,y$ such that
$x \equiv a \pmod{m}$
$y \equiv b \pmod{m}$
$(x,y)=1$?
Equivalently, are there integers $c,d,k,l$ such that $ c(a+mk) + d(b+ml)=1? $
EDIT: Consider two refinements.
1. What conditions on $a,b$ are necessary?
2. What if $m$ is prime?