 Solution

The integer congruence $x\equiv y(\mathrm{mod}n)$ means $n\mid (x-y)$: the difference is an integer multiple of the positive modulus $n$. Write $x_1-x_2=na$ and $y_1-y_2=nb$ for integers $a,b$. Thenso addition preserves integer congruence. For multiplication,The final coefficient is an integer, so multiplication also preserves integer congruence. These properties make addition and multiplication well defined on residue classes.