Suppose I have an linear congruence
$axt \equiv x (\text{mod } m)$ where $gcd(ax, m) = x$.
Then this has $x$ solutions of the form
$t_0 + k(m/x)$ for $k \in \{0, 1, ..., x-1\}$, where $t_0 \equiv a^{-1}(\text{mod }m/x)$.
I think at least one of these solutions is co-prime to $m$, but I am unable to prove that. Does anyone have any idea on how to go about proving that, or is there a paper that contains such a proof (or counter example).
Thanks,
Mike