= Solution
The <integer congruence> $x\equiv y\pmod n$ 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> $a,b$. Then
$$
(x_1+y_1)-(x_2+y_2)=n(a+b),
$$
so <addition> preserves <integer congruence>. For <multiplication>,
$$
x_1y_1-x_2y_2=y_1(x_1-x_2)+x_2(y_1-y_2)=n(ay_1+bx_2).
$$
The final coefficient is an <integer>, so <multiplication> also preserves <integer congruence>. These properties make <addition> and <multiplication> well defined on <residue classes>.
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