Definition

The p-adic valuation or order of a number $n\in\mathbb{N}$, denoted as $v_{p}(n)$ is the highest power of a prime number $p$ that divides $n$ or $$v_{p}(n)=\begin{cases} 0&p \not\mid n \\ \alpha_{i}&p_{i}^{\alpha_{i}}|n \ \text{ and }p_{i}^{\alpha_{i}+1}\not\mid n \end{cases}$$