Why does ab = nc +1 mean the same as ab $\equiv$ 1 (mod n)?

Question

Why does ab = nc +1 mean the same as ab $\equiv$ 1 (mod n)? Is it enough to say that since ab = nc + 1 then ab mod n = 1? Or do we need further steps to clarify?

Answer 1

$ab\equiv 1\pmod n$ means that $ab-1$ is a multiple of $n$, that is, for some integer $c$, we have $ab-1=nc$. Putting $1$ to the other side gives $ab=nc+1$.

Answer 2

because $n\equiv0\pmod n$ so $nc\equiv0\pmod n$ so $nc+1\equiv1\pmod n$