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I can see some solutions online that the answer should be 59, for example in these two links.

  1. https://answers.yahoo.com/question/index?qid=20080910182343AAkvJW0

  2. http://help-notes-22.appspot.com/2012/04/find-the-multiplication-inverses-of-the-given-elementsa-14-in-z15b-38-in-z83

However, I am using an algorithm from my textbook (Abstract Algebra by Beachy and Blair) which I think should follow like the solution below. However, I get answer -24. Is it correct? I know that there are many (infinitely many?) solutions to the equation considering all linear combinations...so which one is the correct multiplicative inverse?

My method:

Using the Euclidean Algorithm (this is the only method for solving that I understand well), \begin{align*} \begin{bmatrix} 1 & 0 & 83\\ 0 & 1 & 38\\ \end{bmatrix} \sim \begin{bmatrix} 1 & -2 & 7\\ 0 & 1 & 38\\ \end{bmatrix} \tag{row 1 minus 2 $\cdot$ (row2)} \\ \sim \begin{bmatrix} 1 & -2 & 7\\ -5 & 11 & 3\\ \end{bmatrix} \tag{row 2 minus 5 $\cdot$(row 1)} \\ \sim \begin{bmatrix} 11 & -24 & 1\\ -5 & 11 & 3\\ \end{bmatrix} \tag{row 1 minus 2 $\cdot ($row 2)}\\ \sim \begin{bmatrix} 11 & -24 & 1\\ -38 & 83 & 0\\ \end{bmatrix}\tag{row 2 minus 3 $\cdot$ (row 1)} \\ \end{align*} Thus $83(11)+38(-24)=1$, which shows that $[38]_{83}^{-1}=-24$.

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    both linked answers are $59$, which is the same as $-24$ i$\mathbb{Z}_{83}$. – 2017-02-09

3 Answers 3

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As you guess, there are "infinitely many" solutions to finding a multiplicative inverse to a given number, modulo 83. That is, there are infinitely many different integers $n$ so that $38n \equiv 1 \pmod{83}$. However, these solutions are not really different, because they're all congruent mod 83. The "correct" multiplicative inverse is the one between 0 and 83, but it's not really more correct than other solutions.

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    Thanks to all the solutions here - I chose this as best answer because of the snippet of info about the "correct" answer being between 0 and 83. It helps me understand the reason why my solution was the minority even if it is technically correct. – 2017-02-09
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It's easy to check, just work out $38\times(-24)$ and $38\times57$, simplify modulo $83$ and see whether or not you get $1$.

Answer: $-24$.

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In the links the answer is $59.$ Your answer $-24$ is correct since $-24 \equiv 59$ mod $83.$ In fact, $-24 +83k$ is a solution for any $k \in \mathbb{Z}.$