The congruence $ax \equiv 1 \mod 1$

Question

Consider this statement:

$ax \equiv 1 \pmod n\,$ has a solution iff $\gcd(a,n) = 1$.

Let $a = 1, n = 1$. Then $\gcd(1,1) = 1$ but no solution exists.

Answer 1

Consider the fact that, for instance, $6\equiv 4\pmod 2$ (i.e. modulo is not an operation in this context). Then apply that thinking to modulo $1$ and see that your problem isn't a problem: $ax\equiv 1\pmod 1$ comes automatically.

Answer 2

I don't see your problem: $x=1$ is a valid solution: $1=1 \pmod{1}$..? Even $x=2$ is one, etc. as $1 \equiv 2 \pmod{1}$ too.