Show that the equation
$$n (a + b) x ≡ [a^2− b^2] \pmod {(a + b)}$$
has solutions.
How many solutions does this equation have?
For the equation to have solutions we need $g= \gcd((a+b), n(a+b))$ to divide $[a^2 - b^2]$.
We have
$$ \begin{align} g & = \gcd(a+b, n(a+b) \\ \\ & = \left | (a+b) \right | \times \gcd(n,1) \\ \\ &= \left | (a+b) \right | \end{align}$$
Since $[a^2 - b^2] = (a+b)(a-b)$, we can see that $g \mid [a^2 - b^2]$ and so the equation does have solutions.
The number of incongruent solutions is the gcd $g$, which is $\left | (a+b) \right |$.