Bunuel wrote:

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.

Kudos for a correct solution.

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MANHATTAN GMAT OFFICIAL SOLUTION:Factors are integers, by definition, so the smallest possible difference between any two factors has to be at least 1. For example, if n = 2, then the number has factors 1 and 2, and the smallest positive difference between those factors is 1. If, on the other hand, n =3, then the number has factors 1 and 3, and the smallest positive difference between those two factors is 2.

On this problem, statement 2 is (arguably) easier, so you might choose to start there.

(2) NOT SUFFICIENT. If n is a multiple of both 11 and 9, then it could be 99. In this case, the factors would be 1, 9, 11, and 99, and the smallest difference between two factors would be 2. On the other hand, n could be 198, with factors 1 and 2 (among others). In this case, the smallest difference is only 1.

(1) SUFFICIENT. What can this strange expression indicate about the value of n? We’re going to need to dig into number theory a bit here.

If that whole expression represents a positive integer, then squaring it would represent a perfect square of an integer:

\(\frac{n+1}{100} = perfect \ square\)

Use the variable p to represent the perfect square, just to make this easier to write:

\(\frac{n+1}{100}=p\)

\(n+1=100p\)

\(n=100p-1\)

\(n=(10\sqrt{p}+1)(10\sqrt{p}-1)\)

Remember that p is a perfect square, so the square root of p is still an integer. This last equation means that one factor of n is \(10\sqrt{p}+1\) and another factor of n is \(10\sqrt{p}-1\) (where \(10\sqrt{p}\) is an integer). These two factors, then, are really “an integer + 1” and “that same integer – 1.” In other words, these two integers are 2 units apart.

But is that the smallest possible distance between two factors? Here’s the best (and trickiest) part. Remember this stage of the equation simplification above?

n=100p-1

That step means: n equals an even number minus 1. In other words, n is odd!

The only way that two factors can be a distance of just 1 unit apart is when one of those factors is even and one of those factors is odd. If n itself is odd, though, then it cannot have any even factors.

Because n is odd, it isn’t possible for two of the factors to be just 1 unit apart. Therefore, the smallest possible distance between two factors is indeed 2.

The correct answer is A.

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