GMAT Practice Question: 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?
- \frac{1}{4}
- \frac{56}{27}
- 2
- 3
- 4
Topics: number theory, divisibility, fractions
Solution
Step 1
We list all positive divisors of 28 by checking which numbers divide 28 without a remainder.
\text{Positive factors of }28 = 1, 2, 4, 7, 14, 28
Step 2
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}
Step 3
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}
Step 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
Answer
C
