Legendre's Formula
Legendre's Formula states that
![\[e_p(n)=\sum_{i\geq 1} \left\lfloor \dfrac{n}{p^i}\right\rfloor =\frac{n-S_{p}(n)}{p-1}\]](http://latex.artofproblemsolving.com/7/a/e/7aec74a5d87d77aaead655f9dc973202626f8093.png)
where 
is the exponent of 
in the prime factorization of 
and 
is the sum of the digits of 
when written in base .
Proof
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