Given n= p^x q^y r^z s^k where p,q,r,s are distinct primes and x.y.z.k not equal to 0 find the smallest possible value of n if it has exactly 12321 positive factors?
n= p^x.q^y.r^z.s^k
n will have its least value when p.q.r.s are the 4 smallest possible primes.
i.e. p = 2, q = 3, r = 5, s = 7. (2 is the smallest prime, and the only even prime. 1 is not a prime)
p^1 is one factor
p^2 is two factors
etc.
p^m is m factors.
From which it follows,
p^x.q^y.r^z.s^k is (x + y + z + k) factors
and, x + y + z k = 12321
The value of n is the least when the smallest prime has the greatest number of multiples, or factors, and the higher primes have lesser multiples.
This gives,
n = 2^(12318).3^1.5^1.7^1
n = 105*2^(12318)