A Perfect Question!

A natural number n n is called perfect when the sum of all its (positive) divisors lower than n n itself equals n n .

Is 6 6 a perfect number?

Not at all! Pheraphs! Can't determine, no matter how hard you try! Yes, of course!

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3 solutions

Andrea Palma
Mar 8, 2016

The divisors of 6 6 , lower than 6 6 itself are 3 3 . Namely 1 , 2 1, 2 and 3 3 . Summing these divisors we have 1 + 2 + 3 = 6 1+2+3 = 6 so, by definition, 6 6 is a perfect number . :) Enjoy!

Avner Lim
Mar 13, 2021

All positive divisors of 6 6 that is lower than 6 6 is 1 , 2 , 1, 2, and 3 3 . 1 + 2 + 3 = 6 1 + 2 + 3 = 6 . So 6 6 is a perfect number. (You can also use the formula "If 2 p 1 2^{p-1} is prime, 2 p 1 ( 2 p 1 ) 2^{p-1}(2^{p}-1) is perfect." Since 6 = 2 2 1 ( 2 2 1 ) 6 = 2^{2-1}(2^{2}-1) , 6 6 is a perfect number.

Mohammad Farhat
Sep 1, 2018

6 = ( 2 2 1 ) ( 2 2 1 ) 6= (2^{2-1})(2^2 -1) So it is perfect

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