Determining $\gcd(94, 27)$ I want to determine $\gcd(94, 27)$.  Using the Euclidean algorithm, I got
\begin{align}
94 &= 27 (3) + 13 \\
\implies 27 &= 13 (2) + 1 \\
\implies \;\;2 &= 2 (1)
\end{align}
Does this mean the GCD is $2$?  Clearly $2$ doesn't divide $27$, so what am I doing wrong?

As mentioned in the comments, the last non-zero remainder is the GCD, not the quotient corresponding to the expression with zero remainder.  To highlight, that is the red/boxed number below:
\begin{align}
94 &= 27 (3) + 13 \\
27 &= 13 (2) + \color{red}{\boxed{1}} \\
13 &= 13  (1) + 0
\end{align}
For another example, consider $\gcd(45, 81)$:
\begin{align}
81 &= (1)45 + 36 \\
45 &= (1)36 + \color{red}{\boxed{9}}  \\
36 &= (4)9  + 0
\end{align}
