Sum of Reciprocals of Divisors of a Perfect Number

If  
\[n\]
  is a perfect number then and  
\[d\]
  is a divisor of  
\[n\]
  then  
\[\sum_{d | n} d = 2n\]
.
we can write this as  
\[\sum_{d | n} \frac{n}{d}=2n\]
 because as  
\[d\]
  runs through all the divisors of  
\[n\]
  so does  
\[\frac{n}{d}\]
. Hence, dividing the last expression by  
\[n\]
  gives  
\[\sum_{d | n} \frac{n/d}{n}=\frac{2n}{n} \rightarrow \sum_{d | n} \frac{1}{d} = 2\]
.

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