Chapter 4.4, Problem 29WE

To determine

To show:the given positive integer is perfect.

Explanation of Solution

Given information:

The given integer:

  $496$

Formula used:

A positive integer n is perfect if it is the sum of all its positive factors except n itself.

Proof:

As per problem,

The integer $496$ can be factored over the positive integers in the following ways:

  $(1,496),\text{ (2,248), (4,124), (8,62), and (16,31)}$

Adding all the positive factors except $496$

  $\begin{array}{l}s=1+2+4+8+16+31+62+124+248\\ =496\end{array}$

As the sum of all the positive factors of $496$ , except $496$ itself, is equal to the integer.

Thus, the integer $496$ is perfect.