# Trailing Zeros: Legendre's Formula ## Illustration With \( n! \) $$ n! = \prod_{p \le n} p^{\alpha(p)} $$ where $$ \alpha(p) = \sum_{m=1}^{\log_p n} \left\lfloor \frac{n}{p^m} \right\rfloor. $$ ## Example for \( n = 10 \) \begin{align*} \alpha(2) &= \left\lfloor \frac{10}{2} \right\rfloor + \left\lfloor \frac{10}{4} \right\rfloor + \left\lfloor \frac{10}{8} \right\rfloor = 5 + 2 + 1 = 8. \\ \\ \alpha(5) &= \left\lfloor \frac{10}{5} \right\rfloor = 2. \end{align*}