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If both \(m\) and \(n\) are positive integers, is \(m\) a multiple of \(n\) ?
(1) \(m + 3n\) is a multiple of \(n\)
(2) \(2m + n\) is a multiple of \(n\)
(1) \(m + 3n\) is a multiple of \(n\)
(2) \(2m + n\) is a multiple of \(n\)
08 Nov 2021, 06:12
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Official Solution:
If both \(m\) and \(n\) are positive integers, is \(m\) a multiple of \(n\) ?
(1) \(m + 3n\) is a multiple of \(n\)
\(3n\) is obviously a multiple of \(n\), thus for the sum of \(m\) and \(3n\) to be a multiple of \(n\), \(m\) must also be a multiple of \(n\). Sufficient.
(2) \(2m + n\) is a multiple of \(n\)
\(n\) is obviously a multiple of \(n\), thus for the sum of \(2m\) and \(n\) to be a multiple of \(n\), \(2m\) must also be a multiple of \(n\). Does \(2m\) being a multiple of \(n\) imply that \(m\) is a multiple of \(n\) ? Not necessarily! For example, if \(m=3\) and \(n=2\), then the answer is NO but if \(m=2\) and \(n=2\), then the answer is YES. Not sufficient.
Answer: A
If both \(m\) and \(n\) are positive integers, is \(m\) a multiple of \(n\) ?
(1) \(m + 3n\) is a multiple of \(n\)
\(3n\) is obviously a multiple of \(n\), thus for the sum of \(m\) and \(3n\) to be a multiple of \(n\), \(m\) must also be a multiple of \(n\). Sufficient.
(2) \(2m + n\) is a multiple of \(n\)
\(n\) is obviously a multiple of \(n\), thus for the sum of \(2m\) and \(n\) to be a multiple of \(n\), \(2m\) must also be a multiple of \(n\). Does \(2m\) being a multiple of \(n\) imply that \(m\) is a multiple of \(n\) ? Not necessarily! For example, if \(m=3\) and \(n=2\), then the answer is NO but if \(m=2\) and \(n=2\), then the answer is YES. Not sufficient.
Answer: A
