If \(n\) is a positive integer greater than \(1\), what is the smallest positive difference...

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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) \(\dfrac{\sqrt{n+1}}{10}\) is a positive integer.

2) \(n\) is a multiple of both \(11\) and \(9\).

The OA is A

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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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