<s>[INST] Solution congruence system $ x \equiv 11\pmod{36},\,x \equiv 7\pmod{40}, \,x \equiv 32\pmod{75}$ Have solution the following congruence system? $$\begin{align}
      x & \equiv 11\pmod{36}\\
      x & \equiv  7\pmod{40}\\
      x & \equiv  32\pmod{75}
    \end{align}$$
Point of Interest: This question requires some special handling due to the mixture of factors among the moduli. This is more than the run of the mill Chinese Remainder Theorem problem.
 [/INST] Hint: It might be easier to break things down into unique and shared factors:
$$
\begin{align}
x&\equiv7\pmod{8}\\
x&\equiv2\pmod{9}\\
x&\equiv7\pmod{25}
\end{align}
$$
Once you have it in this form, you can then use the Extended Euclidean Algorithm to solve
$$
\begin{align}
a&\equiv1\pmod{8}\\
a&\equiv0\pmod{9\cdot25}\\
\end{align}
$$
$$
\begin{align}
b&\equiv1\pmod{9}\\
b&\equiv0\pmod{8\cdot25}\\
\end{align}
$$
$$
\begin{align}
c&\equiv1\pmod{25}\\
c&\equiv0\pmod{8\cdot9}\\
\end{align}
$$
and get
$$
x\equiv7a+2b+7c\pmod{8\cdot9\cdot25}
$$
</s>