With all symbols representing integers, I know:
$$ a \equiv m_a \bmod d_a \\ b \equiv m_b \bmod d_b $$
And I am now looking for $m_c$ and $d_c$ such that:
$$ c = ab \\ c \equiv m_c \bmod d_c \\ \forall d_c', m_c': c \equiv m_c' \bmod d_c' \Rightarrow d_c' \le d_c $$
In words, given two integers for which I know the modulo given a certain divisor, I'm looking for the largest divisor for which I can also know the modulo of their multiplication.
I've had a hard time trying to tackle this analytically. Trying this on some examples I feel like there's some underlying pattern that can yield me a result which is different from $d_c=1$, but I couldn't manage to put my finger on it. Is there a solution different from 1, and if there is, how can I calculate it?