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16 Sep 2014, 01:45
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D
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Difficulty:
95%
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Question Stats:
47% (02:54) correct
53%
(02:33)
wrong
based on 318
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The function \(f(x)\) is defined for all positive integers \(x\) as the number of even factors of \(x\) and the function \(g(x)\) is defined for all positive integers \(x\) as the number of odd factors of \(x\). For positive integers \(a\) and \(b\) if \(f(b)*g(a) = 0\) and \(f(a) = 1\), which of the following could be the least common multiple of \(a\) and \(b\)?
A. 12
B. 16
C. 20
D. 30
E. 36
A. 12
B. 16
C. 20
D. 30
E. 36
16 Sep 2014, 01:45
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Official Solution:
The function \(f(x)\) is defined for all positive integers \(x\) as the number of even factors of \(x\) and the function \(g(x)\) is defined for all positive integers \(x\) as the number of odd factors of \(x\). For positive integers \(a\) and \(b\) if \(f(b)*g(a) = 0\) and \(f(a) = 1\), which of the following could be the least common multiple of \(a\) and \(b\)?
A. 12
B. 16
C. 20
D. 30
E. 36
\(f(b)*g(a) = 0\): Every positive integer has at least one odd factor: 1. Therefore, \(g(a)\) cannot be 0, which implies that \(f(b) = 0\). This, in turn, means that \(b\) is an odd integer (odd integers do not have even factors).
\(f(a) = 1\): \(a\) has one even factor. This means that \(a\) must be 2 (the only positive integer with only one even factor is 2).
The least common multiple of \(a=2\) and \(b=odd\) is \(2*odd\). Only option D could be represented this way: \(30=2*15\).
Answer: D
The function \(f(x)\) is defined for all positive integers \(x\) as the number of even factors of \(x\) and the function \(g(x)\) is defined for all positive integers \(x\) as the number of odd factors of \(x\). For positive integers \(a\) and \(b\) if \(f(b)*g(a) = 0\) and \(f(a) = 1\), which of the following could be the least common multiple of \(a\) and \(b\)?
A. 12
B. 16
C. 20
D. 30
E. 36
\(f(b)*g(a) = 0\): Every positive integer has at least one odd factor: 1. Therefore, \(g(a)\) cannot be 0, which implies that \(f(b) = 0\). This, in turn, means that \(b\) is an odd integer (odd integers do not have even factors).
\(f(a) = 1\): \(a\) has one even factor. This means that \(a\) must be 2 (the only positive integer with only one even factor is 2).
The least common multiple of \(a=2\) and \(b=odd\) is \(2*odd\). Only option D could be represented this way: \(30=2*15\).
Answer: D
General Discussion
30 Sep 2014, 16:17
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Good question Bunuel - Definitely makes you work your logic...actually I kind of used my Critical Reasoning skills for this question. As R.Purewal rightly says - CR is not something that can be learned, it could only be thought through....
