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## Homework Statement

Let's say I want to find the inverse of [itex]\bar{4}[/itex] in [itex]\mathbb{Z}_{13}[/itex].

So I get [itex]13 = 4\cdot 3 + 1[/itex] and so [itex]1 = 13 - 4\cdot 3[/itex].

But this doesn't show that [itex]3[/itex] is inverse of [itex]4[/itex]. So I have to express [itex]4 = 3\cdot 1 + 1[/itex]

which yields that [itex]1 = 4 - 1\cdot 3 = 4 - 3\cdot (13 - 3\cdot 4) = 10\cdot 4 - 3 \cdot 13[/itex] from where I get that [itex]\bar{10}[/itex] is inverse of [itex]\bar{4}[/itex] mod [itex]13[/itex].

So which is the right way for finding inverses in Zn? I'm attaching a screenshot from my book

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