Is there a simple function for combining two residues? Say we have a natural number $x_1$ modulo a natural number $m_1$, and another natural $x_2$ modulo a natural $m_2$.  I'm looking for a function that combines the values so that we get the result of using the Chinese remainder theorem on the numbers.  In other words, I'm looking for a formula that would combine results modulo 3 and modulo 5 into results modulo 15, or modulo $3^2 \cdot 5$ and modulo $5 \cdot 7$ into modulo $3^2 \cdot 5 \cdot 7$.  I'm really looking for a general method to find functions devoted to specific moduli.
It's a major bonus if this formula is easy to incorporate into a summation, but that may be a tough order.

Suppose $x \equiv x_1 \mod m_1$ and $x \equiv x_2 \mod m_2$, where $m_1$ and $m_2$ are coprime.  Let $y$ be the multiplicative inverse of $m_1$ mod $m_2$, 
i.e. $m_1 y \equiv 1 \mod m_2$.  Then $x \equiv x_1 + (x_2 - x_1) m_1 y \mod m_1 m_2$.
