'Distributive' property for a function mod m What properties must some function $f(n)$ have for it to be the case that: $f(n) = (n + 3) \mod m = (n \mod m) + (3 \mod m)$?
Similarly, what if $f(n) = (n + 3) \mod m = (n \mod m + 3)?$
Is this something that is well studied?  Where might I go to find more information?
Suppose here that $n,m \in \mathbb{Z^+}$ $-$ {$0$}, that the equation holds for all or some subset of $m,n$ and that 'mod' stands for the standard modular arithmetic operator.  

Let's assume that $m$ is a positive integer and look at your first equation.
Your equation $(n+3)\mod m=(n\mod m)+(3\mod m)$ is true for all $n$ if $m$ divides $3$: i.e. $m$ is $1$ or $3$. In those cases, both sides of the equation are the same as $n\mod m$: adding the $3$ does nothing.
In all other cases, your equation is not true for all $n$. For $m=2$, use $n=1$. For $m>3$, use $m-3$. In both cases, the left hand side will be zero while the right hand side will be the sum of two non-zero positive integers.
If you want me to answer the second question you will need to clarify your question, as I explain in my comments. To get more information, look at just about any book on number theory.
