<s>[INST] 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
$$
\begin{split}
M &= 88 = 8 \times 11 \\
x_1 &= 123^{456}\equiv 2^{456} \equiv 2^{6} \equiv 64 \equiv 9 \pmod{11} \\
y_1 &= 9^{-1} \equiv 9^9 \equiv (-2)^9 \equiv -512 \equiv -6 \equiv 5 \pmod{11}\\
x_2 &= 123^{456} \equiv 123^0 \equiv 1 \pmod{8}\\
y_2 &= 1^{-1} \equiv 1 \pmod{8} \\
123^{456}
 &\equiv \sum_{i=1}^2 x_i\times\frac{M}{m_i} \times y_i
  \equiv 9\times\frac{88}{11}\times5 + 1\times\frac{88}{8} \times1 \equiv 371 
  \equiv 19 \pmod{88}
\end{split}
$$
 [/INST] You used the inverse of $x_i$ instead of the inverse of $\frac{M}{m_i}$. So for example,
$$
9 \cdot \frac{88}{11} \cdot 5 + 1 \cdot \frac{88}{8} \cdot 1 \equiv 0 + 1 \cdot 11 \cdot 1 \equiv 3 \not \equiv 1\pmod{8}.
$$
If you use the inverse of $\frac{M}{m_i}$ instead, you would have
$$
x_1 \cdot \frac{88}{11} \cdot 8^{-1} + x_2 \cdot \frac{88}{8} \cdot 11^{-1} \equiv 0 + 1 \cdot 11 \cdot 11^{-1} \equiv 1\pmod{8}.
$$
</s>