BTGmoderatorLU wrote: ↑Sun Feb 16, 2020 6:21 am
Source: Manhattan Prep
If \(n\) is a positive integer greater than \(1\), what is the smallest positive difference between two different factors of \(n\)?
1) \(\dfrac{\sqrt{n+1}}{10}\) is a positive integer.
2) \(n\) is a multiple of both \(11\) and \(9\).
The OA is
A
Let's take each statement one by one.
1) \(\dfrac{\sqrt{n+1}}{10}\) is a positive integer.
Say \(\dfrac{\sqrt{n+1}}{10}=x\), a positive integer
\(\dfrac{n+1}{100}=x^2\)
\(n=100x^2-1\)
\(n=(10x–1)(10x+1)\)
We see that 10x is even; thus, (10x – 1) and (10x + 1) are odd. Moreover, \((10x–1)\) and \((10x+1)\) are consecutive odd numbers, which have a difference of 2. Thus, the smallest positive difference between two different factors of \(n\) is 2. Sufficient.
2) \(n\) is a multiple of both \(11\) and \(9\).
Case 1: \(n\) is even: n must have factors 1 and 2. The smallest positive difference between two different factors of n = 2 – 1 = 1.
Case 2: \(n\) is odd: we saw in Statement 1 that if n is odd, one of the possible values of the smallest positive difference between two different factors of \(n\) is 2.
No unique answer. Insufficient.
The correct answer:
A
Hope this helps!
-Jay
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