In this paper we study vp(n!), the greatest power of prime p in factorization of n!. We find some lower and upper bounds for vp(n!), and we show that vp(n!) = (n/p−1) + O(ln n). By using above mentioned bounds, we study the equation vp(n!) = v for a fixed positive integer v. Also, we study the triangle inequality about vp(n!), and show that the inequality pvp(n!) > qvq(n!) holds for primes p < q and sufficiently large values of n.