Question: If x is a positive integer, what is the value of √(x+24) − √x?

(1) √x is an integer
(2) √(x+24) is an integer

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

“If x is a positive integer, what is the value of √(x+24) − √x” is a topic of the GMAT Quantitative reasoning section of GMAT. This GMAT Data Sufficiency question has been taken from the book "501 GMAT Questions". The questions of GMAT Data Sufficiency consist of a problem statement and two factual statements. The quantitative section of the GMAT exam checks the candidates’ knowledge of calculations to solve quantitative problems. The candidates usually overlook the clever accent of words stated in the question which is considered the most difficult portion of these questions. GMAT Quant section includes a series of 31 questions. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach Solution 1:

The problem statement states that

Given:

• x is a positive integer

It is required to find out the value of √(x+24) - √x.

1. The statement expresses that √x is an integer.
   By utilising the facts provided in statement one, we could get numerous values for the equation: √(x+24) - √x.
   If x= 1, then we get:
   √(x+24) - √x = √25 - √1 = 5 - 1 = 4
   If x = 4, then we get:
   √(x+24) - √x = √28 - √4 = √28 - 2.
   Therefore, the statement alone is not sufficient to find out the value of √(x+24) - √x.
2. The statement implies that √(x+24) is an integer.
   By utilising the facts provided in statement two, we could get numerous values for the equation: √(x+24) - √x.
   If x = 1, then we get:
   √(x+24) - √x = √25 - √1 = 5 - 1 = 4;
   If x = 12, then we get:
   √(x+24) - √x = √36 - √12 = 6 - √12.
   Therefore, the statement alone is not sufficient to find out the value of √(x+24) - √x.

Combining statements (1) and (2) together we get:

By utilising the facts provided in statements one and two, we could get numerous values for the equation: √(x+24) - √x.
If x = 1, then we get:
√(x+24) - √x = √25 - √1 = 5 - 1 = 4;
If x = 25, then we get:
√(x+24) - √x = √49 - √25 = 7 - 5 = 2.

Therefore, we are not getting an exact value of the equation (x+24) - √x. Thus statements (1) and (2) together are not sufficient.

Correct Answer: (E)

Approach Solution 2:

The problem statement states that

Given:

• x is a positive integer

Asked:

• To find out the value of √(x+24) - √x.

1. The statement says that √x is an integer.
   Therefore x could have multiple values such as 1, 4,9,25,36,49..any sort of number which must be a perfect square.
   Hence statement (1) alone is insufficient to find out the value of √(x+24) - √x.
2. The statement indicates that √(x+24) is an integer.
   Therefore, x could have multiple values such as 1, 12,25… and so on, so that sum of x and 24 would be a perfect square number.
   Hence statement (2) alone is insufficient to find out the value of √(x+24) - √x.

Combining both statements, we get,

The value of x could be 1, which gives the result 5-1 = 4
Or, the value of x could be 25, which gives the result 7-5= 2
Thus statements (1) and (2) together are not sufficient.

Correct Answer: (E)

Approach Solution 3:

The problem statement states that

Given:

• x is a positive integer

Asked:

• To find out the value of √(x+24) - √x.

1. The statement (1) declares that √x is an integer.
   Since no further information is provided, it can be inferred that x can be any number which will be a perfect square number.
   Therefore, no specific value of x can be determined from this statement.
   Hence the statement alone is insufficient to find the value of √(x+24) - √x.
2. The statement demonstrates that √(x+24) is an integer.
   Similarly, there is no information given in the statement. Therefore, x could have multiple values which will make x+24 a perfect square number.
   Thus, any definite value of x cannot be determined from this statement.
   Therefore, the statement alone is not sufficient to find the value of √(x+24) - √x.

Combining both the statement:
Let’s assume √(x+24)=a and √x=b

Therefore, we get, x+24=a^2 and x=b^2
Thus by putting x=b^2 in the equation x+24=a^2, we get:
=> b^2+24=a^2
=> a^2-b^2=24
=> (a+b)(a-b)=24 (as per the formula of a^2-b^2)

Therefore the possible solutions are:

1. If a+b=12 and a-b=2, then, a=7,b=5
2. If a+b=6 and a-b=4, then a=5 and b=1

Thus we are not getting any specific values of a and b. Thus both statements together are not sufficient to find the value of √(x+24) - √x.

Correct Answer: (E)

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