Can we say that $a=b \implies a \equiv b \pmod{0}$?

Question

I'm wondering if we can write $a=b$ as $a \equiv b \pmod{0}$.

Because the last congruence satisfies $b-a=0\times k$ $\implies b-a=0$.

Which is really $b=a$.

Answer 1

$a \equiv b$ modulo $c$ means that $a - b$ is divisible by $c$, i.e. there exists $k \in \mathbb{Z} $ such that $kc = (a-b)$. So if $c = 0$, $kc = 0$ for all $k$ so $a \equiv b$ modulo $0$ iff $a = b$. So yes. But it's unusual to work modulo $0$, and not very useful, as we see.

Answer 2

Well the notation $a\equiv b\pmod{c}$ means that $a-b$ is divisible by $c$,clearly you can't divide by $0$ though $b-a$ can be written as an linear combination of $0$