Modular Arithmetic/Euler's Theorem

If ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are positive coprime integers, then,

${\displaystyle \alpha ^{\phi (\beta )}\equiv 1{\pmod {\beta }}}$

Where ${\displaystyle \phi }$ denotes Euler's totient function. Here, ${\displaystyle \phi (\beta )}$ gives the number of positive integers up to ${\displaystyle \beta }$ that are relatively prime to ${\displaystyle \beta }$.