Give an example of a linear congruence $ax ≡ b \pmod n$ where integer $d > 1$ divides $a$, $n$, and $b$ but the equation has no solutions.
Since $d > 1$ divides $a$, $n$, and $b$, we have for some integers $p, q, r$
$$ \begin{align} a & = pd \\ \\ b & = qd \\ \\ n & = rd \end{align} $$
So we can rewrite the linear congruence as
$$ pdx \equiv qd \pmod {rd}$$
which reduces to
$$ \begin{align} px & \equiv q \pmod {\frac{rd}{\gcd(rd,d)}} \\ \\ px & \equiv q \pmod {\frac{rd}{d\gcd(r,1)}} \\ \\ & \equiv q \pmod {\frac{rd}{d}} \\ \\ px & \equiv q \pmod {r} \end{align} $$
For there to be no solutions
$$ \gcd(r,p) \not \mid q $$
The candidates $r=2, p=2, q=3$ satisfy $\gcd(r,p) \not \mid q$.
We can then try $d=2$, giving $a=4, b=6, n=4$.
So an example of a linear congruence is
$$ 4x \equiv 6 \pmod 4 $$
Let's check: $g = \gcd(4,4)=4$ and $g \not \mid 6$, so there are no solutions, yet $d=2$ divides 4 and 6.