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09 Jun 2015, 07:59
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
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Question Stats:
66% (01:57) correct
34%
(02:07)
wrong
based on 134
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If \(\frac{a}{x}\) is an integer, where \(a = 5^{15} - 625^3\) and \(x\) is a positive integer greater than 1 that 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
A. None
B. One
C. Two
D. Three
E. Four
09 Jun 2015, 08:00
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Official Solution:
If \(\frac{a}{x}\) is an integer, where \(a = 5^{15} - 625^3\) and \(x\) is a positive integer greater than 1 that 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
First of all, the condition that \(x\) is a positive integer that does not have a factor \(p\) where \(1 < p < x\) effectively means that \(x\) is a prime number.
Moving on, we can rewrite \(a\) as follows: \(a = 5^{15} - 625^3=5^{15} - 5^{12}=5^{12}(5^3-1)=5^{12}*124=2^2*5^{12}*31\).
Finally, considering that \(\frac{a}{x}\) must be an integer and \(x\) is a prime, it becomes clear that \(x\) can have three possible values: 2, 5, or 31.
Answer: D
If \(\frac{a}{x}\) is an integer, where \(a = 5^{15} - 625^3\) and \(x\) is a positive integer greater than 1 that 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
First of all, the condition that \(x\) is a positive integer that does not have a factor \(p\) where \(1 < p < x\) effectively means that \(x\) is a prime number.
Moving on, we can rewrite \(a\) as follows: \(a = 5^{15} - 625^3=5^{15} - 5^{12}=5^{12}(5^3-1)=5^{12}*124=2^2*5^{12}*31\).
Finally, considering that \(\frac{a}{x}\) must be an integer and \(x\) is a prime, it becomes clear that \(x\) can have three possible values: 2, 5, or 31.
Answer: D
19 Nov 2017, 02:46
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abhinashgc
A prime number is a positive integer with exactly two factors: 1 and itself. If an integer does not have a factor which is greater than 1 and less than itself, then it must be a prime. For example, 5 (a prime number) does not have a factor which is more than 1 and less than 5 but 4 has such factor: 2.
