Knowing that every even perfect number is of the form

makes it easy to list all its factors, and this enables us to prove some interesting results. Since is a prime, the factors of must either be a power of (up to ) or a product of a power of and . Listing these factors in a table we get:

Factors of | Other factors |

... | ... |

For example, the factors of are:

Factors of | Other factors |

We claimed that the product of the factors of an even perfect number is always equal to . For this is clearly true, since

as required.

For a general proof, note that the product of the factors in the first column of the table above is

(1) |

As we already noted in the main article, the sum of the first integers is

so (1) is equal to

Note that every factor in the second column of the table above is times the corresponding factor in the first table. Since there are entries in each column, the product of all the factors in the second column is therefore

The product of all the factors is

This proves our result.

A similar result involves adding up the reciprocals of all the factors of an even perfect number. Taking our example of again we see that

In fact the sum is equal to for every perfect number! As one might expect, the proof proceeds in the same way as above, replacing addition with multiplication. We leave it to you as an exercise.