TITLE: How to get all the factors of a number using its prime factorization?
QUESTION [8 upvotes]: For example, I have the number $420$. This can be broken down into its prime factorization of $$2^2 \times3^1\times5^1\times7^1 = 420 $$
Using $$\prod_{i=1}^r (a_r + 1)$$ where $a$ is the magnitude of the power a prime factor is raised by and $r$ is the number of prime factors. I get $24$ possible factors.
Is there an easy way to iterate through all those $4$ factors to obtain all $24$? I know this can be easily done using a table with numbers with only $2$ factors. But, as this one has $4$ I obviously can't implement the table method. So any general solution to do this? Thanks!

REPLY [9 votes]: By way of example of a table of factors, just to check we're talking about the same thing:
The factors of $8=2^3$ are of course just $2^0, 2^1, 2^2, 2^3$ or calculated out  $1,2,4,8$.
The factors of $9=3^2$ are  $3^0, 3^1, 3^2$ aka $1,3,9$.
Now the factors of $72$ can be made up by combining two numbers chosen, one from each of these two sets:
\begin{array}{c|ccc}
\times & 1 & 2 & 4 & 8 \\\hline
1 & 1 & 2 & 4 & 8 \\
3 & 3 & 6 & 12 & 24 \\
9 & 9 & 18 & 36 & 72 \\
\end{array}
This is no doubt the kind of table you are thinking of.
For $420$ we could apply this process repeatedly, just feeding the contents of each table into one axis of the next table:
\begin{array}{c|ccc}
\times & 1 & 2 & 4  \\\hline
1 & 1 & 2 & 4  \\
3 & 3 & 6 & 12 \\
\end{array}
\begin{array}{c|ccc}
\times & 1 & 2 & 4   & 3 & 6 & 12 \\\hline
1 & 1 & 2 & 4  & 3 & 6 & 12 \\
5 & 5 & 10 & 20 & 15 & 30 & 60  \\
\end{array}
\begin{array}{c|ccc}
\times & 1 & 2 & 4  & 3 & 6 & 12 & 5 & 10 & 20 & 15 & 30 & 60 \\\hline
1 & 1 & 2 & 4 & 3 & 6 & 12 & 5 & 10 & 20 & 15 & 30 & 60\\
7 & 7 & 14 & 28 & 21 & 42 & 84 & 35 & 70 & 140 & 105 & 210 & 420\\
\end{array}
or for a slightly more attractive final table we could split the factors into more equal subsets, here the factors of $2^23^1 =12$ across the top and the factors of $5^17^1=35$ down the side.
\begin{array}{c|ccc}
\times & 1 & 2 & 4  & 3 & 6 & 12 \\\hline
1 & 1 & 2 & 4 & 3 & 6 & 12 \\
5 & 5 & 10 & 20 & 15 & 30 & 60\\
7 & 7 & 14 & 28 & 21 & 42 & 84 \\
35 & 35 & 70 & 140 & 105 & 210 & 420\\
\end{array}