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15 Jul 2013, 08:40
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
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If n is a positive integer greater than 1, what is the smallest positive difference between two different factors of n?
(1) \(\frac{\sqrt{n+1}}{10}\) is a positive integer.
(2) n is a multiple of both 11 and 9.
My question is: how would you rephrase the STEM?
a) Since the smallest positive difference between any two different factors of n is n. Would you rephrase the question as:
If n can be any one of 2,3, 4..., what is the value of n ?
or
b) Would you go one step ahead and say what is the smallest possible value of n ?
(1) \(\frac{\sqrt{n+1}}{10}\) is a positive integer.
(2) n is a multiple of both 11 and 9.
My question is: how would you rephrase the STEM?
a) Since the smallest positive difference between any two different factors of n is n. Would you rephrase the question as:
If n can be any one of 2,3, 4..., what is the value of n ?
or
b) Would you go one step ahead and say what is the smallest possible value of n ?
15 Jul 2013, 09:06
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AbhiJ
Your conclusion in (a) is not correct.
Generally the smallest positive difference between two different factors of some integers is greater than or equal to 1. For example:
If n=2, then the smallest positive difference between two different factors of n is 1 --> 2-1=1.
If n=3, then the smallest positive difference between two different factors of n is 2 --> 3-1=2.
If n=4, then the smallest positive difference between two different factors of n is 1 --> 2-1=1.
If n=5, then the smallest positive difference between two different factors of n is 4 --> 5-1=4.
If n=6, then the smallest positive difference between two different factors of n is 1 --> 2-1=1.
If n=7, then the smallest positive difference between two different factors of n is 6 --> 7-1=6.
...
If n=9, then the smallest positive difference between two different factors of n is 2 --> 3-1=2.
...
If n=15, then the smallest positive difference between two different factors of n is 2 --> 3-1=2.
...
Basically if n=even, then the smallest positive difference between two different factors of n is always 1.
If n is odd, then the smallest positive difference between two different factors of n is even, so greater than 1.
Does this make sense?
01 May 2015, 08:41
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I say the answer is A.
(I) indicates that there is some positive integer \(X\) such that \(\frac{\sqrt{n+1}}{10}=X\) --> \(\sqrt{n+1}=10X\) --> \(n+1 = 100X^2\) --> \(n = 100X^2 - 1\) --> \(n = (10X)^2-1^2\) --> \(n = (10X-1)(10X+1)\), and so the smallest difference between any two distinct factors of \(n\) will always be at least \(2\).
Additionally, \(n\) will always be odd because \(10X\) will be even, so \(10X-1\) will be \(odd\), and \(odd*odd = odd\). Because \(n\) is \(odd\), the smallest difference cannot be \(1\), which would imply \(even*odd\).
Therefore, the smallest difference is \(2\), and so (I) is sufficient.
(II) is not sufficient because \(n\) could be \(9*10*11=990\).
(I) indicates that there is some positive integer \(X\) such that \(\frac{\sqrt{n+1}}{10}=X\) --> \(\sqrt{n+1}=10X\) --> \(n+1 = 100X^2\) --> \(n = 100X^2 - 1\) --> \(n = (10X)^2-1^2\) --> \(n = (10X-1)(10X+1)\), and so the smallest difference between any two distinct factors of \(n\) will always be at least \(2\).
Additionally, \(n\) will always be odd because \(10X\) will be even, so \(10X-1\) will be \(odd\), and \(odd*odd = odd\). Because \(n\) is \(odd\), the smallest difference cannot be \(1\), which would imply \(even*odd\).
Therefore, the smallest difference is \(2\), and so (I) is sufficient.
(II) is not sufficient because \(n\) could be \(9*10*11=990\).
