A positive integer n is a perfect number provided that the sum of all the positive factors of...

Question

A positive integer n is a perfect number provided that the sum of all the positive factors of n, including 1 and n, is equal to 2 n. What is the sum of the reciprocals of all the positive factors of the perfect number 28?

GMAT Panda · Accepted Answer

C We list all positive divisors of 28 by checking which numbers divide 28 without a remainder. Positive factors of 28 = 1, 2, 4, 7, 14, 28 We form the sum of the reciprocals of these factors. 1/1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28 We convert the terms with denominators 7, 14, and 28 to a common denominator of 28. 1/7 = 4/28 1/14 = 2/28 1/28 = 1/28 1/7 + 1/14 + 1/28 = 4 + 2 + 1/28 = 7/28 = 1/4 We add this result to the remaining reciprocals. 1 + 1/2 + 1/4 + 1/4 = 1 + 1/2 + 1/2 = 2

Explanation	Calculations	Help
$We list all positive divisors of 28 by checking which numbers divide 28 without a remainder.$	$Positive factors of 28 = 1, 2, 4, 7, 14, 28$
$We form the sum of the reciprocals of these factors.$	$\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}$
$We convert the terms with denominators 7, 14, and 28 to a common denominator of 28.$	$\frac{1}{7} = \frac{4}{28} \frac{1}{14} = \frac{2}{28} \frac{1}{28} = \frac{1}{28} \frac{1}{7} + \frac{1}{14} + \frac{1}{28} = \frac{4 + 2 + 1}{28} = \frac{7}{28} = \frac{1}{4}$
$We add this result to the remaining reciprocals.$	$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{2} + \frac{1}{2} = 2$

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