Is n an Integer? GMAT Problem Solving

Question: Is n an integer?

(1) \(n^2\) is an integer.
(2) \(\sqrt{n}\) is an integer.

Statement 1 alone is sufficient, but statement 2 alone is not sufficient
Statement 2 alone is sufficient, but statement 1 alone is not sufficient
Statement 1 and 2 both are sufficient
Statement 1 and 2 both are insufficient
Both statements together are sufficient but neither is sufficient alone

Correct Answer: B

Solution and Explanation:
Approach Solution 1:
It is asked to determine whether or not n is an integer.
Follow the steps to find that
Let us see statement 1
\(n^2\)is an integer.
If \(n^2\)is an integer, n could be an integer or not. For example, if \(n^2\)= 4, then n is an integer (because n = 2 or -2). If \(n^2\)= 5, however, n is not an integer (because n =\(\sqrt{5}\) or \(\sqrt{-5}\)). The first statement is insufficient to respond to the questions.
Let us see statement 2
\(\sqrt{n}\) is an integer.
n must be an integer in order for \(\sqrt{n}\)to be an integer. This is due to the fact that n = (√n)², and any integer squared is also an integer. The second statement is sufficient to answer the question.
Therefore, Statement 2 alone is sufficient, but statement 1 alone is not sufficient

Approach Solution 2:
There is another approach to solve this question which is pretty simple and could help in finding the answer more quickly, let us look at the reasoning below-
Statement 1
\(n^2\) is an integer. However, n can be or cannot be an integer.
Because
if \(n^2\) = 4 it is an integer
If \(n^2\)= 3 it's not an integer
As a result, Statement 1 alone is insufficient.
Statement 2
\(\sqrt{n}\) is an integer, then n must also be an integer.
Because
\(\sqrt{n}\)= integer
n = \((integer)^2\)
Therefore it can be inferred that Statement 2 alone is sufficient.
Hence, Statement 2 alone is sufficient, but statement 1 alone is not sufficient

Approach Solution 3:
The problem statement asked to determine whether n is an integer or not.

Statement 1: n^2 = Integer
That is we can say n^2 = 1, 2, 3, 4, 5 .… and so on.
i.e. the value of n =1, √2, √3, 2, √5 .… nd so on.
Therefore, we can say whether n is an integer or not. Hence statement one alone is not sufficient.

Statement 2: √n is an integer.
Therefore, we can say, √n= 1, 2, 3, 4, 5 … and so on.
i.e. we can say, n = 1, 4, 9, 16, 25...
Therefore, we can derive that the value of n is an integer.
Hence statement two alone is sufficient.

“Is n an integer”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide 2021”. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions are characterised by a problem statement that is followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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Is n an Integer? GMAT Problem Solving

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