Confusion on how to find answer to $11^{-1}\mod 26$

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I'm confused how to find solutions to questions like $11^{-1}\mod 26$ and others like these.

The solution is $19$ but I don't understand how. $11^{-1}$ on its own is $\frac{1}{11}$.

Thanks

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2 Answers

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In the context of modular arithmetic (and, in general, for abstract algebra), $x^{-1}$ does not mean the reciprocal, necessarily; rather, it means the multiplicative inverse.

That is, $x^{-1}$ is an element such that $xx^{-1}=1$ (where $1$ is whatever multiplicative identity lives in your algebraic universe).

So, to say that modulo $26$, $19=11^{-1}$, really means that $19\cdot11\equiv1\pmod{26}$. (In this case, note that $19\cdot11=209=26\cdot8+1$, so that $19\cdot11\equiv1\pmod{26}$.

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$$11\cdot 7=77=26\cdot 3-1=-1\pmod{26}\implies 11^{-1}=-7=19\pmod{26}$$

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