I'm performing integer arithmetic modulo a number. I'm curious to know what happens if I take the result modulo a different number.
For example, $(10 \bmod 10) \bmod 100 \equiv 0$ and $(10 \bmod 100) \bmod 10 \equiv 0$.
As seen above, I've reversed the congruences. I'd like to know what properties hold, if any, when I switch congruences. The first property I've noticed is that $(a \bmod a\cdot b) \bmod a\cdot c \equiv (a \bmod a\cdot c) \bmod a \cdot b$.
Are there some simple generalized rules for this? I'd like to try to encompass the results of this switching of congruences in as little rules as possible, yet be able to describe as much behavior as possible. Can you please help?