How many solutions are there to $ad\equiv bc \bmod p$

Question

Suppose that $0\leq a,b,c,d\leq p-1$, where $p$ is a prime. Then how many solution are there to $ad\equiv bc \bmod p$?

We can work out the case $p=2$. Suppose that $p>2$.

My approach is this:

1. Case I: One of $a$ or $b$ is $0$. WLOG assume $a=0$. Then $bc \equiv 0 \bmod p$ and hence $b$ or $c=0$. There are $2\times p \times 2 \times p=4p^2 $ ways to form $(a,b,c,d)$.
2. Case II: None of $a \text{ or } b$ is $0$. Then $b,c\not = 0$(otherwise, $ad\equiv 0 \bmod p$ which would mean $a$ or $d$ is $0$.

I did not really get any further and just noticed that $ad \equiv ad \equiv da \equiv (p-d)(p-a) \bmod p$ which is fine but I can't say these are the only solutions.

I would appreciate a hint. Thanks.

Answer 1

When at least one of them is zero:

There are $p^2-(p^2-2p+1)=2p-1$ ways to select $a$ and $d$ and $2p-1$ ways to select $b,c$. So there are $(2p-1)^2$ ways, as André points out.

When none are zero there are $(p-1)^3$ ways to select $a,b,c$ and after that we must solve a congrunce $ad\equiv bc$ which amounts to $d\equiv bca^{-1}$. So there is one solution once $a,b,c$ have been chosen.

Hence there are $(2p-1)^2+(p-1)^3$ solutions.

Answer 2

For the nobody equal to $0$ case, $a$, $b$, and $c$ can be chosen arbitrarily from the non-zero elements. For each choice, there is a unique $d$ that works, giving $(p-1)^3$ ways.

Remark: Note that there are $(2p-1)^2$ cases where the two products are congruent to $0$. The $2p$ that you used instead of $2p-1$ double-counts the pair $(a,d)=(0,0)$.