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I know how to get them (extended Euclidean algorithm using the mod and the number you want the inverse of if their gcd is 1, the inverse is the number you get * the one you have in the linear combination), but I don't understand what this means.

I know inverses are important for decryption for example. I just don't quite understand how it works. And why exactly is it that you can only have an inverse when gcd(a,m)=1 ? (a being any number, m being the mod your are using)

SilenceOnTheWire
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    As for what a multiplicative inverse is in the first place., $a$ has a multiplicative inverse $a^{-1}$ modulo $m$ in the event that $a\times a^{-1}\equiv 1\pmod{m}$. For example $5$ is its own multiplicative inverse mod $12$ since $5\times 5 \equiv 25\equiv 25-12-12\equiv 1\pmod{12}$. As for what they are useful for... suppose we know that $5x\equiv 3\pmod{12}$ and we want to know what $x$ is. Well, by multiplying by the multiplicative inverse of $5$ on both sides we get $5^{-1}\times 5x \equiv 5^{-1}\times 3\equiv 5\times 3\equiv 15\equiv 15-12\equiv 3\pmod{12}$ implying $x\equiv 3$ – JMoravitz Nov 12 '19 at 21:08
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    @JMoravitz Not a good dupe target given all that is asked above. – Bill Dubuque Nov 12 '19 at 21:16
  • @bill I obviously disagree. The linked duplicate covers the part of the question that actually requires thought. The remainder of the question can be covered by a single comment or link to a wikipedia article. We explicitly discourage multiple questions in a single post for exactly this reason, since candidates might answer only half but not all. I picked a candidate that answered the only half that requires a full answer. – JMoravitz Nov 12 '19 at 21:34
  • @JMoravitz If you think that "links to Wikipedia articles" count as answers then we have very different standards for answers. – Bill Dubuque Nov 12 '19 at 21:45
  • Please say something about your background, e.g. have you yet studied congruences or groups or rings? – Bill Dubuque Nov 12 '19 at 21:52
  • @BillDubuque Only congruences. – SilenceOnTheWire Nov 12 '19 at 22:02
  • "Not a good dupe target given all that is asked above." I'm not sure what is being asked. The modulo inverse is self explanitory. The inverse of $k$ modulo $n$ is the value of $x$ so that solves $kx \equiv 1 \pmod n$. What more can be said? Inverses are important because they can undo and solve equivalences. As for how they work... that's answered in the linked article. – fleablood Nov 12 '19 at 23:21

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