Proofs on some modular arithmetic

If $ a_{1} \equiv a_{2}\ (\textrm{mod}\ m) $ and $ b_{1} \equiv b_{2}\ (\textrm{mod}\ m) $, then


The addition and the subtraction

By the definition of modular congruence, we know that when $ a_{1} \equiv a_{2}\ (\textrm{mod}\ m) $, we have

And similarly,

Let (1) - (2), we have

Since $ c_{1} - c_{2} $ must be an integer, by the defination, we get

Similarly, let (1) + (2), we will prove that

The multiplication

According to (1) and (2), we can say



Since, $ c_1,\ c_2,\ m $ are all integers, the coefficient of $(c_{1} c_{2}m + a_2 c_2 + b_2 c_1) m $ must be an integer. By the definition,