Arithmetic Modulo $N$

Suppose $a,\,a',b\,b'\in\mathbb{Z}$ and

$$a\equiv a'\pmod{n}, \qquad b\equiv b'\pmod{n}.$$

Then

$$a + b \equiv a' + b'\pmod{n}$$ (1)

$$a\times b \equiv a'\times b'\pmod{n}$$ (2)

So it makes sense to define $+$ and $\times$ by $[a]+[b]=[a+b]$ and $[a]\times[b]=[a\times b]$.