Last visit was: 23 Jun 2024, 05:21 It is currently 23 Jun 2024, 05:21
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.

# What is the value of positive integer x?

SORT BY:
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 93963
Own Kudos [?]: 634200 [17]
Given Kudos: 82420
Senior Manager
Joined: 02 Dec 2014
Posts: 304
Own Kudos [?]: 303 [5]
Given Kudos: 353
Location: Russian Federation
Concentration: General Management, Economics
GMAT 1: 640 Q44 V33
WE:Sales (Telecommunications)
General Discussion
Manager
Joined: 27 Aug 2014
Posts: 247
Own Kudos [?]: 428 [1]
Given Kudos: 76
Location: Netherlands
Concentration: Finance, Strategy
Schools: ISB '21 LBS '22
GPA: 3.9
WE:Analyst (Energy and Utilities)
Manager
Joined: 27 Jun 2014
Posts: 60
Own Kudos [?]: 113 [1]
Given Kudos: 125
Location: New Zealand
Concentration: Strategy, General Management
GMAT 1: 710 Q43 V45
GRE 1: Q161 V163

GRE 2: Q159 V166
GPA: 3.6
WE:Editorial and Writing (Computer Software)
Re: What is the value of positive integer x? [#permalink]
1
Bookmarks
santorasantu wrote:
Bunuel wrote:
What is the value of positive integer x?

(1) The sum of all unique factors of x is 31
(2) x = y^2, where y is an integer

Kudos for a correct solution.

I think the answer is E:

My approach

from 1: sum of all unique factors of number x = 31. This is an odd number. so, the number should be a perfect square. trying out the perfect sqaured numbers we have:
1, 4, 9, 16, 25, 49 ..

sum of factors for 1: 1
sum of factors for 4: 7
sum of factors for 9: 13
sum of factors for 16: 1+2+4+8+16 = 31 -> possible number
sum of factors for 25: 1+5+25 = 31 --> possible number , already 2 possible solutions so NSF

from 2: given that x is a perfect sqaure, same info as from the stem --> NSF

combined we have 2 answers so NSF
E?

Though E is the correct answer, your analysis of statement 1 is faulty. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is not always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50.

From statement 2 we know that it is perfect square, which isn't sufficient. When we combine, we know it is a perfect square and it's factors add to 31, which gives us two answers as you've rightly mentioned.
Manager
Joined: 27 Aug 2014
Posts: 247
Own Kudos [?]: 428 [0]
Given Kudos: 76
Location: Netherlands
Concentration: Finance, Strategy
Schools: ISB '21 LBS '22
GPA: 3.9
WE:Analyst (Energy and Utilities)
Re: What is the value of positive integer x? [#permalink]
aviram wrote:
santorasantu wrote:
Bunuel wrote:
What is the value of positive integer x?

(1) The sum of all unique factors of x is 31
(2) x = y^2, where y is an integer

Kudos for a correct solution.

I think the answer is E:

My approach

from 1: sum of all unique factors of number x = 31. This is an odd number. so, the number should be a perfect square. trying out the perfect sqaured numbers we have:
1, 4, 9, 16, 25, 49 ..

sum of factors for 1: 1
sum of factors for 4: 7
sum of factors for 9: 13
sum of factors for 16: 1+2+4+8+16 = 31 -> possible number
sum of factors for 25: 1+5+25 = 31 --> possible number , already 2 possible solutions so NSF

from 2: given that x is a perfect sqaure, same info as from the stem --> NSF

combined we have 2 answers so NSF
E?

Though E is the correct answer, your analysis of statement 1 is faulty. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is not always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50.

From statement 2 we know that it is perfect square, which isn't sufficient. When we combine, we know it is a perfect square and it's factors add to 31, which gives us two answers as you've rightly mentioned.

True you are correct, I purposefully picked only the perfect squares as to make the solution more concise.I edited my first statement.
Thanks for the point.
Math Expert
Joined: 02 Sep 2009
Posts: 93963
Own Kudos [?]: 634200 [0]
Given Kudos: 82420
Re: What is the value of positive integer x? [#permalink]
Bunuel wrote:
What is the value of positive integer x?

(1) The sum of all unique factors of x is 31
(2) x = y^2, where y is an integer

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION

Solution: E.

Statement 2 is clearly not sufficient as plenty of numbers are squares of integers (1, 4, 9, 16, 25, 36, etc.). When testing statement 1, then, it's a good idea to test candidates that will work with statement 2, as well. 25 is a good option: the factors of 25 are 1, 5, and 25 for a sum of 31. x could be 25. The only other number in that list with a chance is 16. And the factors of 16 are 1, 2, 4, 8, and 16. That sum is also 31. Because statements 1 and 2 each allow for two different values of x, the informations is not sufficient and the correct answer is E.
Senior Manager
Joined: 29 Jun 2017
Posts: 312
Own Kudos [?]: 811 [0]
Given Kudos: 76
GPA: 4
WE:Engineering (Transportation)
Re: What is the value of positive integer x? [#permalink]
1) SUM OF ALL UNIQUE FACTORS = 31
X= 17x11x2x1 => sum =31
X= 17x13x1 => sum = 31
2) X=Y^2 , Y = integer
X=1 , Y=1
X=4, Y=2
X=9, Y=3
B eliminated

Combine 1&2

X= Y^2
Y= 17x13x1 => X= (17x13x1)^2
Y= 17x11x2x1 => X= (17x11x2x1)^2
C is eliminated

Ans is E
VP
Joined: 10 Jul 2019
Posts: 1387
Own Kudos [?]: 562 [0]
Given Kudos: 1656
Re: What is the value of positive integer x? [#permalink]
Concept: to find the SUM of all the unique positive factors of a given value, you can find all the unique combinations of the prime bases of the number and then using common factoring end up with the following Rule/Formula:

If a number - N’s prime factorization is given by:

N = (p)^2 * (q)^1

Then for each unique prime, you add up each Prime from the 0th power to its highest power

And then take the product of each one of the SUMS you find (as illustrated below for the hypothetical value of N above):

SUM of unique positive factors of N = (p^0 + p^1 + p^2) * (q^0 + q^1)

What is the value of integer X = ?

Statement 1: the sum of all the factors of X is 31

Since 31 is a prime number itself, using the above concept to find the SUM of all the unique positive factors, you can infer that there must be only ONE UNIQUE PRIME FACTOR which divides X.

This is because the only 2 positive integers that multiply to 31 are——-> (1) * (31) = 31 ——> and since we include (prime)^0 = 1 in each one of our “factors”, we could never have such a scenario.

For example: if there were 2 prime factors that divided X, such that —- N = (p) (q)

then the SUM of the factors would be:

(p^0 + p^1) * (q^0 + q^1) = 31

We could never have one of the Binomial Factors of the above product be just 1 since any base to the 0th power = 1 already.

Thus If the SUM of all the unique positive factors of X is 31, then we can Infer that X is made up of ONLY 1 Unique Prime Factor.

Case 1: try a value of X composed of only the prime factor 2

If X = (2)^4

Then the sum of all the unique prime factors would be:

(2^0 + 2^1 + 2^2 + 2^3 + 2^4) =

(1 + 2 + 4 + 8 + 16) = 31

X can = 16

Case 2: can we use the prime factor of 3 and get the same result?

3’0 = 1
3’1 = 3
3’2 = 9
3’3 = 27 ———-> too high, will not work

Case 3: X is composed only of the Prime Factor 5

If X = (5)^2

Then the SUM of all the unique prime factors would be:

(5^0 + 5^1 + 5^2) =

(1 + 5 + 25) = 31

X can = 25 as well

Since we have two different values for X, statement 1 not sufficient

*note* X can not = 31 because both 1 and 31 divide into 31 and the sum of the factors would be 32, not 31, in violation of statement 1

S2: X = (Y)^2

All statement 2 tells us is that X must be a perfect square.

Thus X = 16 and X = 25 both work.

Together: since case X can = 16 or 25, the statements will be not sufficient together either.

E

Posted from my mobile device
Non-Human User
Joined: 09 Sep 2013
Posts: 33675
Own Kudos [?]: 840 [0]
Given Kudos: 0
Re: What is the value of positive integer x? [#permalink]
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
Re: What is the value of positive integer x? [#permalink]
Moderator:
Math Expert
93963 posts