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I know how to use the Euclidean algorithm to find the inverse modulo in most cases, but I can't wrap my head around the calculations when the modulo is smaller than the number I'd like to find the inverse for.

For example:

$$59x \equiv 1 \pmod{19} $$

has solution $$x \equiv 10 \pmod{19}$$ according to online calculators but I can't figure out why.

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    $\begingroup$ 59 and 1 are not congruent mod 19! $\endgroup$ – user21820 Dec 12 '14 at 13:07
  • $\begingroup$ @user21820 : What does that mean? If you check the link you can see the calculation I'd like to find the steps for. $\endgroup$ – OptimusCrime Dec 12 '14 at 13:09
  • $\begingroup$ You wrote $59 \equiv 1 \pmod{19}$. That is simply false! See my answer for what you should write. $\endgroup$ – user21820 Dec 12 '14 at 13:10
  • $\begingroup$ It just means that $59\cdot 10\equiv 1\bmod 19$. $\endgroup$ – Dietrich Burde Dec 12 '14 at 13:10
  • $\begingroup$ By the way I didn't downvote your question but I can see why someone might for the reason I take issue with what you wrote. $\endgroup$ – user21820 Dec 12 '14 at 13:13
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What you want is a solution to $59x \equiv 1 \pmod{19}$.

But $59x \equiv 1 \pmod{19} \Leftrightarrow (3(19)+2)x \equiv 1 \pmod{19} \Leftrightarrow 2x \equiv 1 \pmod{19}$.

I'm sure you can now solve the last equation, which is equivalent to the original.

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  • $\begingroup$ Thanks. I'll accept as soon as I can. What about $$19x \equiv 1 \pmod{59} $$? This method does not work then. $\endgroup$ – OptimusCrime Dec 12 '14 at 13:39
  • $\begingroup$ What method are you referring to? The algebraic manipulations I showed above is to solve your question of what to do when you want to find the inverse of $x$ mod $m$ where $|x| > m$. In any case, once you find that $59 \cdot 10 \equiv 1 \pmod{19}$, you have $59 \cdot 10 - 19k = 1$ for some integer $k$ and hence you also have found an inverse of 19 mod 59 by definition of inverse. $\endgroup$ – user21820 Dec 12 '14 at 14:01
  • $\begingroup$ @OptimusCrime: Anyway if you find any answer on Math SE helpful, not just to your questions, you can upvote it. $\endgroup$ – user21820 Dec 12 '14 at 14:03

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