# Thread: exponent of a prime number in a factorial..!!

1. ## exponent of a prime number in a factorial..!!

I was trying to deduct the logic of computing the exponent of a prime number p in n!, I could not get the logic at the first place, and then when I tried to go through some books, they referred to [n] where the [ ] denotes greatest integer <= n.

Any help or explanation in this matter.

DOTMATRIX

2. Originally Posted by dotmatrix
I was trying to deduct the logic of computing the exponent of a prime number p in n!, I could not get the logic at the first place, and then when I tried to go through some books, they referred to [n] where the [ ] denotes greatest integer <= n.

Any help or explanation in this matter.

DOTMATRIX
I assume you're talking about the fact that if $\displaystyle \eta(n,p)=\max\{k^k\mid n\}$ then $\displaystyle \displaystyle \eta(n,p)=\sum_{k=1}^{\infty}\left\lfloor\frac{n}{ p^k}\right\rfloor$?

3. [duplicate]

For a prime $\displaystyle p$, in $\displaystyle n!$,
each number $\displaystyle 1 < kp \leq n$ contributes 1 to the exponent,
and each number $\displaystyle 1 < kp^2 \leq n$ contributes another 1 to the exponent,
and each number $\displaystyle 1 < kp^3 \leq n$ contributes another 1 to the exponent,
...

From this, exponent of p in n! :
$\displaystyle \displaystyle \sum_{m=1}^\infty \lfloor \frac{n}{p^m} \rfloor$

4. got it bro..thanx a lot.

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### exponent of prime in n factorial

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