SOLUTION: what is the least positive integer that has exactly 9 factors?

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Question 938445: what is the least positive integer that has exactly 9 factors?
Found 2 solutions by Fombitz, KMST:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you mean unique prime factors (?).
N=2%2A3%2A5%2A7%2A11%2A13%2A17%2A19=9699690
If they don't have to be unique,
N=2%5E9=512
If they don't have to be prime,
N=2%2A3%2A4%2A5%2A6%2A7%2A8%2A9%2A10=3628800

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
A number has more or less factors depending on its prime factorization.
If the prime factorization is
n=A%5Ea%2AB%5Eb%2A%22....%22Y%5Ey%2AZ%5Ez
those prime factors can be arranged into products of the form
A%5E%28alpha%29%2AB%5E%28beta%29%2A%22...%22
with 0%3C=alpha%3C=a , 0%3C=beta%3C=b , etc
to form all factors of n
from 1=A%5E0%2AB%5E0%2A%22...%22 to
n=A%5Ea%2AB%5Eb%2A%22...%22 .
The number of possible products is
%28a%2B1%29%2A%28b%2B1%29%2A%22...%22%2A%28y%2B1%29%2A%28z%2B1%29 ,
with as many such brackets as prime factors,
(multiplied together if there is more than one),
and each bracket being 2 or more.
In this case, 9=3%2A3 is the number of factors,
meaning that there are just 2 prime factors,
both with 2 as an exponent:
n=A%5E2%2AB%5E2 with a=b=2 .
The smallest n we can make with that formula is
n=2%5E2%2A3%5E2=4%2A9=36 .