Prime factorization of n factorial -

how find prime factorization of n! when n large number(10^8)? efficient way this?

start finding primes $n$, perhaps sieve of eratosthenes. mark largest primes having exponent 1, way down (but not including) n/2. mark ones n/2 n/3 having exponent 2, , forth; continue until sqrt(n). @ point might compute each exponent separately, function (pseudocode):

factorial_exponent(n, p):   n <- floor(n/p)   let t = n   while n >= p:     n <- floor(n/p)     t <- t + n   return t 

you use function on primes instead of marking them in ranges suggested, though take little bit longer.