If $a\equiv b\pmod m$ and $c+d\equiv 0\pmod m$ then $ac+bd\equiv 0\pmod m$

Question

If $a\equiv b\pmod m$ and $c+d\equiv 0\pmod m$ then $ac+bd\equiv 0\pmod m$.

The response, posted below is correct??

Answer 1

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If $a\equiv b\pmod m\Longrightarrow m\mid a-b\Longrightarrow m\mid c(a-b)\Longrightarrow m\mid ac-bc\Longrightarrow \\ac-bc=mk$

If $c+d\equiv 0\pmod m\Longrightarrow m\mid c+d\Longrightarrow m\mid b(c+d)\Longrightarrow m\mid bc+bd\\bc+bd=mj$

Adding, $ac-bc+bc+bd=mk+mj\\ac+bd=m(k+j)\\m\mid ac+bd\Longrightarrow ac+bd\equiv0\pmod m\;\;\;\;\;\;\;\Box$