# Math Help - Number of factors

1. ## Number of factors

The number 12 is the smallest number with 6 distinct factors: 1, 2, 3, 4, 6, and 12. Find the smallest number that has 24 distinct factors.

Now, I can do this given some time and trial and error. What I am interested in knowing is if there is a more "constructive" way to do this? I can't seem to think of one.

Thanks!

-Dan

2. Originally Posted by topsquark
The number 12 is the smallest number with 6 distinct factors: 1, 2, 3, 4, 6, and 12. Find the smallest number that has 24 distinct factors.

Now, I can do this given some time and trial and error. What I am interested in knowing is if there is a more "constructive" way to do this? I can't seem to think of one.

Thanks!

-Dan
The numbers you mention are called highly composite numbers.

This link might be of interest: Highly composite number - Wikipedia, the free encyclopedia

And this link has an algorithm: http://wwwhomes.uni-bielefeld.de/ach...anuscript3.pdf

3. Originally Posted by topsquark
The number 12 is the smallest number with 6 distinct factors: 1, 2, 3, 4, 6, and 12. Find the smallest number that has 24 distinct factors.

Now, I can do this given some time and trial and error. What I am interested in knowing is if there is a more "constructive" way to do this? I can't seem to think of one.

Thanks!

-Dan
Consider the prime decomposition of a number $n$:

$
n=p_1^{k_1}...p_m^{k_m}
$

Then the number of factors of $n$ is $\prod_{i=1}^m(k_i+1)$

From here it is easy.

RonL