$D$ is a PID and $a=bc$, a random factorization of some $0\neq a \in D$. I want to prove $\langle b \rangle / \langle a \rangle \simeq D/\langle c \rangle$
The condition $a=bc$ suggest $1/c=b/a$. Maybe define $\phi: \langle b \rangle \to D/\langle c \rangle , \phi(x)= xc-bc$ which is a morphism in which case $\ker \phi = \langle a \rangle$ and $\frac{\langle b \rangle}{\langle \ker \phi \rangle}\simeq \frac{D}{\langle c \rangle}$.