Last visit was: 25 Jul 2024, 01:55 It is currently 25 Jul 2024, 01:55
Toolkit
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

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

SORT BY:
Tags:
Show Tags
Hide Tags
Manager
Joined: 22 Sep 2015
Posts: 73
Own Kudos [?]: 1024 [310]
Given Kudos: 136
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 3710
Own Kudos [?]: 17364 [137]
Given Kudos: 165
Manager
Joined: 23 Apr 2015
Posts: 234
Own Kudos [?]: 521 [122]
Given Kudos: 36
Location: United States
WE:Engineering (Consulting)
General Discussion
Manager
Joined: 03 Jan 2015
Posts: 65
Own Kudos [?]: 305 [1]
Given Kudos: 146
Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
1
Bookmarks
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?
Tutor
Joined: 16 Oct 2010
Posts: 15148
Own Kudos [?]: 66863 [10]
Given Kudos: 436
Location: Pune, India
Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
5
Kudos
5
Bookmarks
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.
Intern
Joined: 09 Aug 2016
Posts: 42
Own Kudos [?]: 66 [0]
Given Kudos: 8
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.
Math Expert
Joined: 02 Sep 2009
Posts: 94614
Own Kudos [?]: 643722 [2]
Given Kudos: 86748
Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
1
Kudos
1
Bookmarks
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.
Board of Directors
Joined: 11 Jun 2011
Status:QA & VA Forum Moderator
Posts: 6047
Own Kudos [?]: 4768 [1]
Given Kudos: 463
Location: India
GPA: 3.5
Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
1
Kudos
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.

Intern
Joined: 16 Sep 2017
Posts: 7
Own Kudos [?]: 7 [3]
Given Kudos: 1
Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
3
Kudos
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
Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 19199
Own Kudos [?]: 22718 [3]
Given Kudos: 286
Location: United States (CA)
Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
1
Kudos
2
Bookmarks
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.)

Intern
Joined: 23 Nov 2018
Posts: 18
Own Kudos [?]: 14 [0]
Given Kudos: 104
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
Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 19199
Own Kudos [?]: 22718 [0]
Given Kudos: 286
Location: United States (CA)
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

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

CEO
Joined: 07 Mar 2019
Posts: 2627
Own Kudos [?]: 1873 [0]
Given Kudos: 763
Location: India
WE:Sales (Energy and Utilities)
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.

Manager
Joined: 19 Dec 2020
Posts: 160
Own Kudos [?]: 41 [0]
Given Kudos: 316
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.
Intern
Joined: 24 Dec 2022
Posts: 5
Own Kudos [?]: [0]
Given Kudos: 118
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?
GMAT Club Legend
Joined: 08 Jul 2010
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Posts: 6031
Own Kudos [?]: 13828 [2]
Given Kudos: 125
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Re: A positive integer n is a perfect number provided that the sum of all [#permalink]
2
Kudos
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  (4.9/5 google rated GMAT Prep Destination)- Book your FREE trial session
Providing Focused GMAT Prep (Online and Offline) for GMAT Focus along with 100% successful Admissions counselling
http://www.Youtube.com/GMATinsight (LIKE and SUBSCRIBE the channel for 1100+ topic-wise sorted Videos)
Get TOPICWISE Self-Paced course: Concept Videos | Practice Qns 100+ | Official Qns 50+ | 100% Video solution CLICK HERE

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?

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