I know how to get a inverse but I don't know how to put it into equations. So let $5^{-1}$ exist in mod m. Is $5^{-1}$, $1/5$, $-5$, or the modular inverse number. To put it more generally, is $5^{-1}$ a notation or really a power of $-1$?
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Welcome to Mathematics Stack Exchange. $-5$ is the additive inverse; the others are the multiplicative inverse. And $5^{-1}$ behaves like a power in that $5^{-1}5^n\equiv5^{n-1}\pmod m$ when $5^{-1}$ exists – J. W. Tanner Aug 12 '21 at 03:37
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Welcome to MSE! For best results I suggest writing in Mathjax. Here is a quick tutorial to get you started. To answer your question, $5^{-1}$ usually denotes the multiplicative inverse of 5 in any number system. – Daniel Aug 12 '21 at 03:38
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It can be either - see the linked dupes for the fractional viewpoint. Recall the modular inverses are unique so there is no inconsistency. – Bill Dubuque Aug 12 '21 at 04:05