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

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 2n. What is the sum of the reciprocals of all the positive factors of the perfect number 28?

A) 1/4
B) 56/27
C) 2
D) 3
E) 4

Can somebody explain what the question asks and gives ? Is quite convoluted I would say.

A) 1/4
B) 56/27
C) 2
D) 3
E) 4

EgmatQuantExpert · Accepted Answer

Here is a methodical approach to solve this question. Given : A positive integer N is considered a perfect number if the sum of its factors is equal to 2N . Eg : 6 is a perfect number since 1 + 2 + 3 + 6 = 12 = 2*6 Given that 28 is a perfect number. Required : Sum of reciprocals of the factors of 28 . Point to keep in mind : Note that every factor of number N is always multiplied with another factor to get N Eg : 1 * 6 = 6 ; 2 * 3 = 6 ; 3 * 2 = 6 ; 6 * 1 = 6 Approach : 1. To calculate the sum of reciprocals, take the common denominator as N itself ( 28 in this case). a. For instance, if we consider 6 as our example number, the sum of reciprocals of factors is \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = \frac{(6 + 3 + 2 + 1)}{6} 2. Since all factors of 28 are distinct, the numerator of each fraction becomes the corresponding “other factor” of N . a. Eg : \frac{1}{2} is same as \frac{3}{6} 3. So, the numerator of the sum is simply the sum of all factors. a. In the case of our example number 6 , you could see that the sum of reciprocals of factors is \frac{(6 + 3 + 2 + 1)}{6} and the numerator is simply the sum of factors of 6 itself. 4. We know the sum of factors of the given number 28 a. It is given that 28 is a perfect number and that the sum of factors of a perfect number is twice the number itself. Working Out : 1. Sum of reciprocals of the factors of 28 = S = \frac{(Sum Of Factors of 28)}{28} 2. S = 2*28/28 = 2 Correct Answer : Option C Note : I haven’t focused on listing down the factors of 28 since that has been done in each of the posts above. Please refer to the previous posts to check the calculations. My primary objective in this post is to highlight the underlying logic in the question: the reason that the question clearly states the definition of a perfect number and explicitly tells us that 28 is a perfect number. Point To Note : Although it is not incorrect to list down the factors of the given number, such an approach is not a scalable approach. For instance, had the question mentioned 496 (which is also a perfect number) instead of 28 , listing down the factors would have been way too cumbersome and some might even begin to deem the question as “unsolvable in 2 minutes”. However, if you're someone who focused on the logic mentioned above, you would immediately identify that even in the case of 496 (or any other perfect number), the sum of reciprocals of all its factors is 2 again. There is a reason that the question gives us the definition (and the property) of a perfect number and tells us explicitly that 28 is a perfect number. The test maker wants to see if the test taker (you) can identify and use the given property of a perfect number combined with a common observation to arrive at the answer elegantly. (The observation is reiterated as a Takeaway below.) Takeaway : For a positive integer N , the sum of reciprocals of factors is simply \frac{(Sum Of Factors Of N)}{N} Footnote : Do you need to remember this result? Not really. As long as you understand how this has been arrived at, you’d come up with many such interesting results while solving any good question using a methodical approach. Hope this helps. Cheers,

Krishna

Senthil1981 · Answer

Answer is C:

Before going to generic result, consider just for 28, where the factors are 1, 2, 4, 7, 14, 28 and sum of these are 2*28.
Therefore the sum of the inverse of the factors are \(\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}\) and since all the denominators are factors of 28, simplifying the above equation will result in
\(\frac{1}{28} * (28+14+7+4+2+1)\)
= \(\frac{1}{28} * (2 * 28)\)
= 2

(Share a kudos. if you like the explanation) :-D

saiesta · Answer

I did it the exact same way, however, I feel that there should be an even easier way to solve this. Any other methods?

KarishmaB · Answer

Use approximation:

1 + 1/2 + 1/4 + ... (some very small numbers)
It will be more than 1 but 27 cannot be in the denominator since there is no multiple of 3.
Answer must be 2.

Ndkms · Answer

Can somebody explain what the question asks and gives ? Is quite convoluted I would say.

Bunuel · Answer

A perfect number is a positive integer if the sum of its factors equals twice that number. For example, 6 is a perfect number because the sum of the factors of 6 (which are 1, 2, 3, and 6) is 6*2 = 12: 1 + 2 +3 + 6 = 12.

We are given another perfect number 28 and asked to find the sum of the reciprocals of its factors, so to find 1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28.

Hope it's clear.

Abhishek009 · Answer

Lets try -

Given

1. The number is 28
2. The sum of all the positive factors of 28 ( Including 1 ) is 2*28 = 56

Work out

We first find all the positive factors of the perfect number 28.

Positive factors of 28 including 1 is  1, 2, 4, 7, 14, 28

Then we are required to find -

The reciprocal of the numbers highlighted above is provided below -

And for solving the question you can adopt this method -

Hope this answers your query..

Anantz · Answer

N is perfect number

Perfect number is 2n of sum of factors reciprocal

28 is perfect number

These are given

Isn't answer straight away 2?

Am I missing something?

Posted from my mobile device

ScottTargetTestPrep · Answer

We are given that 28 is a perfect number. The reason it is a perfect number is because its factors are 1, 2, 4, 7, 14, and 28, which add up to 56, which is twice 28. Now we need to find the sum of the reciprocals of these factors; that is, we need to determine the value of 1/1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28. Since the LCD of all denominators is 28, we have:

28/28 + 14/28 + 7/28 + 4/28 + 2/14 + 1/28

Notice that all the numerators now are the factors of 28, which add up to 56, so the sum is:

56/28 = 2

(Note: In fact, the sum of the reciprocals of all the positive factors of any perfect number is 2. For example, 6 is also a perfect number (6 and 28 are the two smallest perfect numbers). The factors of 6 are 1, 2, 3, and 6, and the sum of their reciprocals is 1/1 + 1/2 + 1/3 + 1/6 = 6/6 + 3/6 + 2/6 + 1/6 = 12/6 = 2.)

Answer: C

legendinthewomb · Answer

You actually don't need to use the first statement.

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 2n. What is the sum of the reciprocals of all the positive factors of the perfect number 28?

A) 1/4
B) 56/27
C) 2
D) 3
E) 4

Can somebody explain what the question asks and gives ? Is quite convoluted I would say.

n: sum of factors=2n
n=28 then sum of 1/factors =?

Factors=1,2,4,7,14, and 28

Sum of reciprocals= 1/1+1/2+1/4+1/7+1/14+1/28
= (28+14+7+4+2+1)/28
=2*28/28 =2

ScottTargetTestPrep · Answer

Solution:

The factors of 28 are 1, 2, 4, 7, 14, and 28 (notice that 28 is a perfect number because 1 + 2 + 4 + 7 + 14 + 28 = 56, which is exactly twice 28). Therefore, the sum of the reciprocals of its factors is:

1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28

28/28 + 14/28 + 7/28 + 4/28 + 2/28 + 1/28

(28 + 14 + 7 + 4 + 2 + 1)/28

56/28 = 2

Answer: C

unraveled · Answer

Factors of 28 are 1, 2, 4, 7, 14 and 28
Sum of Reciprocals  = \frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}  
 = (1 + \frac{1}{7}) + (\frac{1}{2} + \frac{1}{14}) + (\frac{1}{4} + \frac{1}{28}) 
 = (1 + \frac{1}{7}) + \frac{1}{2}(1 + \frac{1}{7}) + \frac{1}{4}(1 + \frac{1}{7}) 
 = (1 + \frac{1}{7}) + (1 + \frac{1}{7})(\frac{1}{2} + \frac{1}{4}) 
 = (1 + \frac{1}{7})(1 + \frac{3}{4}) 
 = \frac{8}{7} * \frac{7}{4} 
= 2

There are many ways to add the fractions. One can start backwards with  \frac{1}{14} + \frac{1}{28}  as well and solve.
Of course the formula is far helpful in higher numbers with more factors and a complex calculation.

Answer C.

ueh55406 · Answer

if  n = 28 , then  2n= 56

meaning the sum of ALL the +ve factors of 28 should be = 56

factors of  28 = 1+2+4+7+14+28 = 56

so, as required in the question :  1/1+1/2+1/4+1/7+1/14+1/28= 2  Answer is C.

TylerFerreira1 · Answer

Hey Bunuel, question - isn't the whole first part irrelevant? Why do we need to know what a "perfect number" is?

GMATinsight · Answer

Hi  TylerFerreira1

A good thing about GMAT is it always gives you the definition of lesser-known terms if the question stem uses such terms. The question here has done the same. For the audience to understand the meaning of the term perfect number the definition is shared by question.

However, your question too is relevant here. There was possibly no need for the definition or term perfect number here and the question alone "What is the sum of the reciprocals of all the positive factors of 28?" would have been sufficient.

GMATinsight · Answer

factors of 28 = {1, 2, 4, 7, 14, 28}

Sum of reciprocal  = (\frac{1}{1}) + (\frac{1}{2}) + (\frac{1}{4}) + (\frac{1}{7}) + (\frac{1}{14}) + (\frac{1}{28}) = (\frac{(28+14+7+4+2+1)}{28}) = (\frac{(56)}{28}) = 2

Answer: Option C

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

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

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