number theory - Any relationship between $amod b$ and $amod c$?

Let's say we have $a\equiv x\mod b$ and $a\equiv y\mod c$. Is there any way to find the relationship between $x$ and $y$ for nontrivial (a>b and/or a>c) moduli?

How about a specific example, such as

$a\equiv x\mod b$ vs $a\equiv y\mod 2b$,

or

$a\equiv x\mod b$ vs $a\equiv y\mod 2b-1$.