Modulo Arithmetic/Examples/Multiplicative Inverse of 41 Modulo 97

Example of Modulo Arithmetic
The inverse of $41$ under multiplication modulo $97$ is given by:
 * ${\eqclass {41} {97} }^{-1} = 71$

Proof
From Ring of Integers Modulo Prime is Field, multiplication modulo $97$ has an inverse for all $x \in \Z_{97}$ where $x \ne 0$.

Using Euclid's Algorithm:

Then:

So:
 * $\paren {-26} \times 41 \equiv 1 \pmod {97}$

Hence the result.