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Im looking at a homework problem I have and I am a bit confused. The first part of the question is to show that 8128 is a perfect number. This is simple enough: $1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128$. The second part confuses me though

What is the highest power of 2 that it is factorable by? Give its cofactor. What can be said of that cofactor?

The highest power of $2$ is $2^6$, this is simple. But what is the cofactor? As far as I remember, cofactors only deal with matrices, which are not part of this problem at all. Is there some other definition I do not know about?

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  • $\begingroup$ If $\ n = a b\,\ $ then $\ b\,$ is the cofactor of $\,a\,$ (in $\,n).\ $ Cofactor means "complementary factor". $\ $ $\endgroup$ – Bill Dubuque Dec 20 '13 at 17:10
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It is an odd use of the term. What is meant here is the quotient $\frac{8128}{2^6}$, that is, $127$. You had found this during your calculation of the sum of the divisors.

Recall that the even perfect numbers are the numbers of the form $2^{p-1}(2^p-1)$ where $2^p-1$ is prime (and therefore so is $p$).

Probably you are expected to note that $127$ is prime, and that it is $2(2^6)-1$.

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  • $\begingroup$ So the cofactor is 127 and we can say that the cofactor is equal to the highest power of 2? Can I also ask where you read this, or perhaps what the better term would be? $\endgroup$ – David says Reinstate Monica Dec 20 '13 at 17:05
  • $\begingroup$ The cofactor is $2M-1$ where $M$ is the highest power of $2$. It is an old result of Euclid that if $2^n-1$ is prime, then $2^{n-1}(2^n-1)$ is perfect. It was shown by Euler that any even perfect number has to be of this shape. The Wikipedia article on perfect numbers will tell you all these things, and much more. $\endgroup$ – André Nicolas Dec 20 '13 at 17:08

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