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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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13
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

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. $$S = 2*28/28 = 2$$

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

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29
6

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) ##### General Discussion
Manager  Joined: 03 Jan 2015
Posts: 72
Re: A positive integer n is a perfect number provided that the sum of all  [#permalink]

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Senthil1981 wrote:

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) 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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3
3
saiesta wrote:
Senthil1981 wrote:

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

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28 = 1*28
2*14
4*7
sum of reciprocals = 1+1/28+1/2+1/14+1/4+1/7=56/28=2
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Re: A positive integer n is a perfect number provided that the sum of all  [#permalink]

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Can somebody explain what the question asks and gives ? Is quite convoluted I would say.
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Joined: 02 Sep 2009
Posts: 59624
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:

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.

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

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

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1
TheLordCommander wrote:

_________________________
Yes.
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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

All-purpose approach

We know that all positive factors of a number $$N=p^a*q^b*r^c…$$ is a geometric progression with the sum:

$$S = \frac{p^{a+1} – 1}{p-1} * \frac{q^{b+1} – 1}{q-1} * \frac{r^{c+1} – 1}{r-1} …$$

Reciprocals of $$N$$ is still a geometric progression with only one distinction – powers are negative in this case.

Back to our question:

$$28 = 2^2*7$$

$$\frac{2^{-3} – 1}{2^{-1} - 1} * \frac{7^{-2} – 1}{7^{-1} - 1} = \frac{7}{4} * \frac{8}{7} = 2$$

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

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3
N is perfect number

Perfect number is 2n of sum of factors reciprocal

28 is perfect number

These are given

Am I missing something?

Posted from my mobile device
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A positive integer n is a perfect number provided that the sum of all  [#permalink]

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Quote:
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?

Anantz wrote:
N is perfect number

Perfect number is 2n of sum of factors reciprocal

28 is perfect number

These are given

Am I missing something?

Posted from my mobile device

Anantz , see bolded text. What is given: sum of all positive factors = 2n.

The prompt does not say that the sum of the factors' reciprocals = 2n

Now I am trying to figure out whether you misread what was given, or you know a simple formula that can be derived from what is given. (I don't think the latter is true. If I am wrong, please correct me.)

Hope that helps.
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genxer123

May be I am wrong. I think any number 'n' whose sum of regular positive factors give us 2n then the reciprocal too would give us similar answer. E. G.

6 = 1, 2,3,6 =2n =12

If I go and take reciprocals 1/1, 1/2, 1/3 & 1/6. Lcm would be 6 and numerator would end up becoming 2 times always of denominator. .

If my generalization is far fetched my bad. Still a learner.

Posted from my mobile device
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A positive integer n is a perfect number provided that the sum of all  [#permalink]

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Anantz wrote:
genxer123

May be I am wrong. I think any number 'n' whose sum of regular positive factors give us 2n then the reciprocal too would give us similar answer. E. G.

6 = 1, 2,3,6 =2n =12

If I go and take reciprocals 1/1, 1/2, 1/3 & 1/6. Lcm would be 6 and numerator would end up becoming 2 times always of denominator. .

If my generalization is far fetched my bad. Still a learner.

Anantz , not far fetched. Quite the opposite: spot on. You are correct. And a brainiac. Kudos.

(I am still a learner, too. I looked it up.)
Quote:
The sum of reciprocals of all the divisors of a perfect number is 2 . . .

That quote is a little bit hard to find. It is the fourth full paragraph from the bottom of the regular text.

Weisstein, Eric W. "Perfect Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PerfectNumber.html
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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.)

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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:

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) can someone explain, how from this $$\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}$$

we get this $$\frac{1}{28} * (28+14+7+4+2+1)$$ please Senior PS Moderator V
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Re: A positive integer n is a perfect number provided that the sum of all  [#permalink]

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dave13 wrote:
Senthil1981 wrote:

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) can someone explain, how from this $$\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}$$

we get this $$\frac{1}{28} * (28+14+7+4+2+1)$$ please Hi dave13

For any fraction: $$\frac{1}{a} + \frac{1}{b} = \frac{1}{LCM(a,b)}*(b + a)$$

Here, consider the fraction $$\frac{1}{4} + \frac{1}{7} = \frac{1}{LCM(4,7)}*(7 + 4) = \frac{1}{28}(7 + 4)$$

Here, we know that
$$28*\frac{1}{28} = 1 | 28*\frac{1}{14} = 2 | 28*\frac{1}{7} = 4 | 28*\frac{1}{4} = 7 | 28*\frac{1}{2} = 14$$

Therefore, $$\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{7} + \frac{1}{14} + \frac{1}{28}$$ = $$\frac{1}{28} * (28+14+7+4+2+1)$$

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

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EgmatQuantExpert wrote:
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

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. $$S = 2*28/28 = 2$$

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

Thanks Krishna for the detailed solution !!
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" The few , the fearless " Re: A positive integer n is a perfect number provided that the sum of all   [#permalink] 11 Apr 2018, 22:34

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