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16 Sep 2014, 01:44
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If \(N = 3^x * 5^y\), where \(x\) and \(y\) are positive integers, and N has 12 positive factors, what is the value of N?
(1) 9 is NOT a factor of N.
(2) 125 is a factor of N.
(1) 9 is NOT a factor of N.
(2) 125 is a factor of N.
16 Sep 2014, 01:44
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Official Solution:
If \(N = 3^x * 5^y\), where \(x\) and \(y\) are positive integers, and N has 12 positive factors, what is the value of N?
Since \(N = 3^x * 5^y\) has 12 positive factors, this means that \((x+1)(y+1)=12=2*6=6*2=3*4=4*3\). We can have the following cases:
\(N = 3^1*5^5\);
\(N = 3^5*5^1\);
\(N = 3^2*5^3\);
\(N = 3^3*5^2\).
(1) 9 is NOT a factor of N.
\(9 = 3^2\) is NOT a factor of N, implies that the power of 3 is less than 2, thus N can only be \(3^1*5^5\). Sufficient.
(2) 125 is a factor of N.
\(125 =5^3\) is a factor of N, implies that the power of 5 is at least 3, thus N could be either \(3^1*5^5\) or \(3^2*5^3\). Not sufficient.
Answer: A
If \(N = 3^x * 5^y\), where \(x\) and \(y\) are positive integers, and N has 12 positive factors, what is the value of N?
Since \(N = 3^x * 5^y\) has 12 positive factors, this means that \((x+1)(y+1)=12=2*6=6*2=3*4=4*3\). We can have the following cases:
\(N = 3^1*5^5\);
\(N = 3^5*5^1\);
\(N = 3^2*5^3\);
\(N = 3^3*5^2\).
(1) 9 is NOT a factor of N.
\(9 = 3^2\) is NOT a factor of N, implies that the power of 3 is less than 2, thus N can only be \(3^1*5^5\). Sufficient.
(2) 125 is a factor of N.
\(125 =5^3\) is a factor of N, implies that the power of 5 is at least 3, thus N could be either \(3^1*5^5\) or \(3^2*5^3\). Not sufficient.
Answer: A
RenanBragion
23 Jun 2020, 05:18
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I think this is a high-quality question and I agree with explanation.
