Question 14
Find the total number of solutions to the following system of equations:
$$
\begin{cases}
a^2 + bc \equiv a \pmod{37}, \\
b(a+d) \equiv b \pmod{37}, \\
c(a+d) \equiv c \pmod{37}, \\
bc + d^2 \equiv d \pmod{37}, \\
ad - bc \equiv 1 \pmod{37}.
\end{cases}
$$
$$
\begin{cases}
a^2 + bc \equiv a \pmod{37}, \\
b(a+d) \equiv b \pmod{37}, \\
c(a+d) \equiv c \pmod{37}, \\
bc + d^2 \equiv d \pmod{37}, \\
ad - bc \equiv 1 \pmod{37}.
\end{cases}
$$