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a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer 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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a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer 18 May 2015, 07:10
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Bunuel
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

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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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a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer 06 Apr 2017, 00:38
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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 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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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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a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer 06 Apr 2017, 00:55
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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 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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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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a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer 02 Mar 2018, 10:44
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Bunuel
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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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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a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer 31 May 2024, 01:36
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Can someone help explain how do we know that x is a prime no. ?

It says that x doesnt have a factor p. If I take x =6, then we can have 1<5<6. here 5 is not a factor of 6 and 6=x is not a prime no.? I am not able to interpret the statement that x is prime no.

Thanks
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a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer 31 May 2024, 01:44
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nikitathegreat
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

Can someone help explain how do we know that x is a prime no. ?

It says that x doesnt have a factor p. If I take x =6, then we can have 1<5<6. here 5 is not a factor of 6 and 6=x is not a prime no.? I am not able to interpret the statement that x is prime no.

Thanks
Show more

6 does have factors p such that 1 < p < 6. For example, both 2 and 3 are factors of 6, satisfying 1 < p < 6. Only prime numbers satisfy the statement because a prime number has exactly two factors: 1 and itself, and does not have any other factor p such that 1 < p < prime.
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a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer 03 Sep 2025, 08:20
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a=5^15 - 5^12

a=5^12 (124)

a=5^12 (2^2 *31)

5,2,31
D
Bunuel
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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a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer 07 Sep 2025, 08:10
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Hope my process is good if anyone have anything easy feel free to response
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a = 5^15 - 625^3 and a/x is an integer, where x is a positive integer 25 Apr 2026, 23:19
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Really silly, but I am not sure why many a times I consider 625 to be 5^3 instead of 5^4.
Apart from that:

5^15 - 5^12:
5^12[5^3 - 1]
= 5^12 * 124.
= 5^12 * 4 * 31.
Prime numbers: 2,5,31.

Answer: Option D
Bunuel
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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