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

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Bunuel
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a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer [#permalink] New post   15 May 2015, 03:50
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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

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Bunuel
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Re: a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer [#permalink] New post   18 May 2015, 07:10
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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.

Answer: D.
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Re: a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer [#permalink] New post   06 Apr 2017, 00:38
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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.

Answer: D.



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
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Re: a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer [#permalink] New post   06 Apr 2017, 00:55
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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.

Answer: D.



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

Answer: D
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Re: a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer [#permalink] New post   27 Sep 2020, 22:28
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a = \(5^{15} - 625^{3}\)

=> a = \(5^{15} - (5^4)^{3}\)

=> a = \(5^{15} - 5^{12}\)

=> a = \(5^{12} [5^3 -1]\)

=> a = \(5^{12}\) * 124

=> a = \(5^{12}\) * 2 * 2 * 31.

Now since \(\frac{a}{x}\) is an integer then x [It is a prime] can be either 2, 5, or 31.

Answer D
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Re: a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer [#permalink] New post   01 Mar 2023, 23:44
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Re: a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer [#permalink]
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