Perfect Number/Examples/496

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Example of Perfect Number

$496$ is a perfect number:

$1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496$


Proof

\(\ds 496\) \(=\) \(\ds 16 \times 31\)
\(\ds \) \(=\) \(\ds 2^{5 - 1} \paren {2^5 - 1}\)

Thus $496$ is in the form $2^{p - 1} \paren {2^p - 1}$.

$2^5 - 1 = 31$ is prime.

So $496$ is perfect by the Theorem of Even Perfect Numbers.


The aliquot parts of $496$ are enumerated at $\sigma_0$ of $496$:

$1, 2, 4, 8, 16, 31, 62, 124, 248$

$\blacksquare$