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Difficulty: 555-605 Level,   Arithmetic,   Number Properties,                           
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
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Senthil1981 wrote:
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

I did it the exact same way, however, I feel that there should be an even easier way to solve this. Any other methods?
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
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saiesta wrote:
Senthil1981 wrote:
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

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


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.
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
Can somebody explain what the question asks and gives ? Is quite convoluted I would say.
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
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Ndkms wrote:
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 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.
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
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Ndkms wrote:
Can somebody explain what the question asks and gives ? Is quite convoluted I would say.

Lets try -

nycgirl212 wrote:
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?


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 -

Senthil1981 wrote:
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


And for solving the question you can adopt this method -
VeritasPrepKarishma wrote:
It will be more than 1 but 27 cannot be in the denominator since there is no multiple of 3.
Answer must be 2.


Hope this answers your query..
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
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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?

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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
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nycgirl212 wrote:
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


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
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]

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?

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


nycgirl212 wrote:
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
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
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nycgirl212 wrote:
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


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
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
nycgirl212 wrote:
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

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.
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
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.
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
Bunuel wrote:
Ndkms wrote:
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 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.

­Hey Bunuel, question - isn't the whole first part irrelevant? Why do we need to know what a "perfect number" is? 
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
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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. :)
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TylerFerreira1 wrote:
Bunuel wrote:
Ndkms wrote:
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 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.

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

­
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Re: A positive integer n is a perfect number provided that the sum of all [#permalink]