TITLE: Euclid Algorithm to Find Muliplicative Inverse
QUESTION [7 upvotes]: Here I am trying to find the multiplicative inverse of 19 respect to 29.
$$19x \equiv 1 \pmod{29} $$
What I tried
\begin{align*}
29 &= 1(19) + 10\\\
19 &= 1(10) + 9\\\
10 &= 1(9) + 1.
\end{align*}
From backtracking, I came up with the
\begin{align*}
1 &= 2(29) - 3(19)\\\
\end{align*}
However, 3 is not a multiplicative inverse of the 29. Where am I making a mistake?
I looked many answers including this answer; however, couldn't figure out my mistake.

REPLY [3 votes]: Reducing your backtracking result modulo $29$, it becomes
$$
1\equiv -3\cdot 19\pmod{29}
$$
Which is to say, the multiplicative inverse of $19$ is $-3$.

REPLY [3 votes]: You're almost there!  Multiply both sides by $-3$ and you have $$-57x\equiv -3\pmod {29}\\x\equiv-3\equiv26\pmod{29}$$