Congruations - relations

Question

Is

$3n+2 \equiv 0 \bmod 4$ same as

$3n-6 \equiv 0 \bmod 4$ ?

I think it's the same thing because $2$ and $-6$ have the same remainder when divided with $4$.

Answer 1

Yes, your argument is correct:\begin{align}3n+2\equiv3n-6\pmod4&\iff(3n+2)-(3n-6)\equiv0\pmod4\\&\iff8\equiv0\pmod4,\end{align}which is true, since $4\mid8$.

Answer 2

In general: $$a\equiv b\mod4\iff4\mid a-b$$

So the statements $3n+2\equiv 0\mod4$ and $3n-6\equiv 0\mod4$ can be translated into $4\mid 3n+2$ and $4\mid 3n-6$.

It is not difficult to prove that these statements are equivalent.

Answer 3

Yes $\!\bmod m\!:\,\ \color{#c00}{y\equiv y'}\ \ \Rightarrow\,\ \ x+\color{#c00}y\equiv x+\color{#c00}{y'}$

because $\,\ \ m\,\mid \color{#c00}{y-y'}\!\Rightarrow m\mid x\!+\!y\!-\!(x\!+\!y') = \color{#c00}{y-y'}$

This law is known as the $ $ Congruence Sum Rule.