Using the Chinese Remainder Theorem, $17x \equiv 9 \pmod{276}$

Solution 1:

Below I show how to easily find the errors. Recall (read!) that the reason the CRT formula works is because each summand has the sought value for one modulus, and is $\equiv 0\,$ for all others. Thus your last summand $\,s = \color{#0a0}{13}\cdot 2\cdot\color{#c00} 5\,$ should satisfy $\,s\equiv 0 $ mod $3\ \&\ 4$, and have the sought value mod $23$, i.e. $\,s\,$ should be a root of $\,17\:\! s\equiv 9\pmod{\!23}$.

But your $\,s\not\equiv 0 $ mod $3\ \&\ 4$. The CRT formula achieves that by including a $\rm\color{#0a0}{first\ factor}$ of $\,3\cdot 4 = 12$, but your first factor is $\color{#0a0}{13}$. Fixing that typo your summand becomes $\,s = 12\cdot 2\cdot\color{#c00} 5$.

Finally $\,s\,$ must be a root of $17s\equiv 9\pmod{23}\,$ but yours has $17s\equiv 15\not\equiv 9$. The CRT formula achieves that by choosing a root $\,r\,$ then writing $\,s = 12\:\!(12^{-1}\bmod 23)\:\!r\equiv r.\,$ Your 2nd factor $\,12^{-1}\equiv 2\,$ is correct but your $\rm\color{#c00}{3rd\ factor}$ $\,r\equiv \color{#c00}5\,$ is not a root since $17\cdot 5\equiv 17\not\equiv 9$. Let's fix that by calculating a root $\,r\,$ by twiddling to exact quotients

$$\bmod 23\!:\,\ 17r\equiv 9\iff r\equiv \dfrac{9}{17}\equiv\dfrac{9}{-6}\equiv\dfrac{-3}{2}\equiv\dfrac{20}2\equiv 10\qquad\qquad$$

Thus the correct summand for modulus $\,23\,$ is $\,s = 12\cdot 2\cdot 10$.

Notice how a good understanding of the reason that the CRT formula works allowed us to easily troubleshoot the problem. This is true in general - if you understand the idea behind a proof or formula then you can debug an erroneous application of it be going through the proof line-by-line to determine the first place where the proof breaks down in your special case. For more examples of this debugging method see a "proof" that $1 = 0$ and a "proof" that $2 = 1$.

Solution 2:

as Peter has said in the comments,
$x\equiv 10\pmod{23}$
the last equation is supposed to be:
$92*2*0 +69*1*1 + 12*2*10=309\equiv 33\pmod{276}$