Finding remainder of $123^{456}$ divided by 88 using Chinese Remainder Theorem I tried using Chinese remainder theorem but I kept getting 19 instead of 9.
Here are my steps
$$
\begin{split}
M &= 88 = 8 \times 11 \\
x_1 &= 123^{456}\equiv 2^{456} \equiv 2^{6} \equiv 64 \equiv 9 \pmod{11} \\
y_1 &= 9^{-1} \equiv 9^9 \equiv (-2)^9 \equiv -512 \equiv -6 \equiv 5 \pmod{11}\\
x_2 &= 123^{456} \equiv 123^0 \equiv 1 \pmod{8}\\
y_2 &= 1^{-1} \equiv 1 \pmod{8} \\
123^{456}
 &\equiv \sum_{i=1}^2 x_i\times\frac{M}{m_i} \times y_i
  \equiv 9\times\frac{88}{11}\times5 + 1\times\frac{88}{8} \times1 \equiv 371 
  \equiv 19 \pmod{88}
\end{split}
$$

$y_1$ should've been the inverse of $8\pmod{11}$, not of $9\pmod{11}$, so $y_1=7$.
Similarly, $y_2$ should've been the inverse of $11\pmod 8$, not of $1\pmod 8$, so $y_2=3$.
Therefore, the result is: $9\times\frac{88}{11}\times \color{red}{7}+1\times\frac{88}{8}\times \color{red}{3}=537\equiv 9\pmod{88}$
