# Thread: Power of P dividing n!

1. ## Power of P dividing n!

If $x>0$ is a real number define $[x]$ to be greatest integer less than or equal to $x.$ If $p$ is a prime, show that the power of $p$ which exactly divides $n!$ is given by $\Bigl[\frac{n}{p}\Bigr] + \Bigl[\frac{n}{p^2}\Bigr]+ \cdots + \Bigl[\frac{n}{p^k}\Bigr]+ \cdots$

2. Originally Posted by Chandru1
If $x>0$ is a real number define $[x]$ to be greatest integer less than or equal to $x.$ If $p$ is a prime, show that the power of $p$ which exactly divides $n!$ is given by $\Bigl[\frac{n}{p}\Bigr] + \Bigl[\frac{n}{p^2}\Bigr]+ \cdots + \Bigl[\frac{n}{p^k}\Bigr]+ \cdots$

Suppose $kp\le n<(k+1)p\,,\,k\in\mathbb{N}$ $\Longrightarrow k\le \frac{n}{p}<\frac{k+1}{k}p\Longrightarrow \left[\frac{n}{p}\right]=k=$ the number of times p enters ONCE in $n\Longrightarrow k=$ the number of times p divides n!

Suppose $mp^2\le n<(m+1)p^2\,,\,m\in\mathbb{N}$ $\Longrightarrow m\le \frac{n}{p^2}<\frac{m+1}{m}p^2\Longrightarrow \left[\frac{n}{p^2}\right]=m=$ the number of times p^2 enters TWICE in $n\Longrightarrow m=$ the number of times $p^2$ divides n!

Etc....

Tonio