Show that the equation
$$ \frac{a} {g} x ≡ b (mod \frac{n}{ g} ) $$
where $g= \gcd (a, n)$ has solutions.
How many solutions does this equation have?
For the equation to have solutions we need $f = \gcd( \frac{n}{g}, \frac{a}{g} )$ to divide $b$.
We have
$$ \begin{align} f & = \gcd( \frac{n}{g}, \frac{a}{g} ) \\ \\ & = 1 \quad \text{Proposition 1.5}\end{align} $$
This divides any integer, including $b$, and so the equation has solutions.
The number of solutions is $f$, that is, there is only one incongruent solution.