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29 Jul 2016, 02:20
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C
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80%
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If \(n = x^5*y^7\), where \(x\) and \(y\) are positive integers greater than 1, how many positive divisors does \(n\) have?
(1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\).
(2) \(n\) has only two prime factors.
(1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\).
(2) \(n\) has only two prime factors.
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17 Jul 2017, 10:28
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Official Solution:
If \(n = x^5*y^7\), where \(x\) and \(y\) are positive integers greater than 1, how many positive factors does \(n\) have?
(1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\).
This statement implies that both \(x\) and \(y\) are primes (a prime number does not have a factor that is greater than 1 and less than itself; it has only two factors: 1 and itself). Now, if \(x\) and \(y\) are different primes, then the number of factors of \(n\) will be \((5+1)(7+1)=48\). However, if \(x=y\), then \(n = x^5*y^7=x^{12}\) and it will have 13 factors. Not sufficient.
(2) \(n\) has only two prime factors.
If those primes are \(x\) and \(y\), then the number of factors of \(n\) will be \((5+1)(7+1)=48\). But if, say, \(x=2*3=6\) and \(y=2\), then \(n = x^5*y^7=2^{12}*3^5\) and it will have \((12+1)(5+1)=78\) factors. Not sufficient.
(1)+(2) Since from (2) \(n\) has only two prime factors, then from (1) it follows that \(x\) and \(y\) are different primes; so the number of factors of \(n\) will be \((5+1)(7+1)=48\). Sufficient.
Answer: C
If \(n = x^5*y^7\), where \(x\) and \(y\) are positive integers greater than 1, how many positive factors does \(n\) have?
(1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\).
This statement implies that both \(x\) and \(y\) are primes (a prime number does not have a factor that is greater than 1 and less than itself; it has only two factors: 1 and itself). Now, if \(x\) and \(y\) are different primes, then the number of factors of \(n\) will be \((5+1)(7+1)=48\). However, if \(x=y\), then \(n = x^5*y^7=x^{12}\) and it will have 13 factors. Not sufficient.
(2) \(n\) has only two prime factors.
If those primes are \(x\) and \(y\), then the number of factors of \(n\) will be \((5+1)(7+1)=48\). But if, say, \(x=2*3=6\) and \(y=2\), then \(n = x^5*y^7=2^{12}*3^5\) and it will have \((12+1)(5+1)=78\) factors. Not sufficient.
(1)+(2) Since from (2) \(n\) has only two prime factors, then from (1) it follows that \(x\) and \(y\) are different primes; so the number of factors of \(n\) will be \((5+1)(7+1)=48\). Sufficient.
Answer: C
General Discussion
unverifiedvoracity
Joined: 12 Jan 2016
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Concentration: Operations, General Management
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WE:Supply Chain Management (Consumer Electronics)
29 Jul 2016, 15:54
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You can calculate the number of divisors of n, if you know that x and y are prime.
(1) x does not have a factor p such that 1 < p < x and y does not have a factor q such that 1 < q < y.
The wording of this statement and the question stem implicitly states that x and y are different integers. Given this statement, x and y are distinct prime numbers, so the number of divisors of n can be calculated using this statement.
This statement is sufficient.
(2) n has only two prime factors.
From the question stem \(n =x^5∗y^7\). This statement cannot be used to ascertain that x and y are the 2 prime factors of n. Let me illustrate with an example -
When \(n = 4^5 * 6^7\) and you are told that n has 2 prime factors, this does not mean 4 and 6 are the two prime factors. The 2 prime factors of n in this case are 2 and 3.
Hence, Statement 2 by itself is not sufficient to answer the question.
My answer is A.
(1) x does not have a factor p such that 1 < p < x and y does not have a factor q such that 1 < q < y.
The wording of this statement and the question stem implicitly states that x and y are different integers. Given this statement, x and y are distinct prime numbers, so the number of divisors of n can be calculated using this statement.
This statement is sufficient.
(2) n has only two prime factors.
From the question stem \(n =x^5∗y^7\). This statement cannot be used to ascertain that x and y are the 2 prime factors of n. Let me illustrate with an example -
When \(n = 4^5 * 6^7\) and you are told that n has 2 prime factors, this does not mean 4 and 6 are the two prime factors. The 2 prime factors of n in this case are 2 and 3.
Hence, Statement 2 by itself is not sufficient to answer the question.
My answer is A.
