# Is there any modular arithmetic property relating $\mod mn$ to $\mod m$?

We know that addition, subtraction and multiplication can be defined for integer modular arithmetic:

for $a \equiv b \mod n$ and $c \equiv d \mod n$, $a+c \equiv b+d \mod n$ and so on.

But is there any modular arithmetic property that relates $a \equiv b \mod m$ and $a \equiv c \mod mn$? Assume that every number here is an integer.

Consider that $a\equiv b (mod m)$ $\rightarrow$ a= b + lm and $a\equiv c (mod mn)$ $\rightarrow$ a= c + kmn . But then l= kn. What's this mean? Here's another hint: This result in number theory is sometimes called "casting out the moduls" or "reducing modulo m". What's this mean? Do you know?