It is currently 17 Mar 2018, 21:09

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer

Author Message
TAGS:

### Hide Tags

Intern
Joined: 20 Sep 2013
Posts: 1
Location: India
Concentration: Entrepreneurship, General Management
WE: General Management (Manufacturing)
Re: a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer [#permalink]

### Show Tags

06 Apr 2017, 00:38
Bunuel wrote:
Bunuel wrote:
a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer such that it does NOT have a factor p such that 1 < p < x, then how many different values for x are possible?

A. None
B. One
C. Two
D. Three
E. Four

Kudos for a correct solution.

OFFICIAL SOLUTION:

First of all, notice that x is a positive integer such that it does NOT have a factor p such that 1 < p < x simply means that x is a prime number.

Next, $$a = 5^{15} - 625^3=5^{15} - 5^{12}=5^{12}(5^3-1)=5^{12}*124=2^2*5^{12}*31$$.

Finally, for a/x to be an integer where x is a prime, x can take 3 values: 2, 5, or 31.

Hi Bunnel,

Can you please clarify if the value of x = 2, then according to 1 < p < x, what will be the value of p. I marked option C because x has to be a prime number and I assumed p = 2 as its given 1 < p < x and x has to be greater than p
Math Expert
Joined: 02 Sep 2009
Posts: 44290
Re: a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer [#permalink]

### Show Tags

06 Apr 2017, 00:55
sujay840 wrote:
Bunuel wrote:
Bunuel wrote:
a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer such that it does NOT have a factor p such that 1 < p < x, then how many different values for x are possible?

A. None
B. One
C. Two
D. Three
E. Four

Kudos for a correct solution.

OFFICIAL SOLUTION:

First of all, notice that x is a positive integer such that it does NOT have a factor p such that 1 < p < x simply means that x is a prime number.

Next, $$a = 5^{15} - 625^3=5^{15} - 5^{12}=5^{12}(5^3-1)=5^{12}*124=2^2*5^{12}*31$$.

Finally, for a/x to be an integer where x is a prime, x can take 3 values: 2, 5, or 31.

Hi Bunnel,

Can you please clarify if the value of x = 2, then according to 1 < p < x, what will be the value of p. I marked option C because x has to be a prime number and I assumed p = 2 as its given 1 < p < x and x has to be greater than p

We are told that x is a positive integer such that it does NOT have a factor p such that 1 < p < x.

2 does not have a factor p such that 1 < p < 2.
_________________
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 2293
Location: United States (CA)
Re: a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer [#permalink]

### Show Tags

02 Mar 2018, 10:44
Bunuel wrote:
a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer greater than 1, such that it does NOT have a factor p such that 1 < p < x, then how many different values for x are possible?

A. None
B. One
C. Two
D. Three
E. Four

We can start by simplifying a:

5^15 - 625^3 = 5^15 - (5^4)^3 = 5^15 - 5^12 = 5^12(5^3 - 1) = 5^12(124) = 5^12(4)(31) = 5^12( 2^2)(31)

If a/x is an integer, then x is a factor of a. However, if x does not have a factor p such that 1 < p < x, then x must be prime number. For example, if x = 5, we see that x doesn’t have a factor between 1 and itself. Since a has three distinct prime factors, there are three distinct values for x: 2, 5 and 31.

_________________

Scott Woodbury-Stewart
Founder and CEO

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Re: a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer   [#permalink] 02 Mar 2018, 10:44

Go to page   Previous    1   2   [ 23 posts ]

Display posts from previous: Sort by