If x is a positive integer, what is the value of (x + 24)^(1/2) - x

We are given that x is a positive integer and must determine the value of √(x+24) - √x.

Statement One Alone:

√x is an integer.

Using the information in statement one, we could obtain multiple values for √(x+24) - √x. If x = 1, then √(x+24) - √x = √25 - √1 = 5 - 1 = 4; however, if x = 4, then √(x+24) - √x = √28 - √4 = √28 - 2. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

√(x+24) is an integer.

Using the information in statement two, we could obtain multiple values for √(x+24) - √x. If x = 1, then √(x+24) - √x = √25 - √1 = 5 - 1 = 4; however, if x = 12, then √(x+24) - √x = √36 - √12 = 6 - √12. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Using the information in statements one and two, we still could obtain multiple values for √(x+24) - √x. If x = 1, then √(x+24) - √x = √25 - √1 = 5 - 1 = 4; however, if x = 25, then √(x+24) - √x = √49 - √25 = 7 - 5 = 2.

Answer: E
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If x is a positive integer, what is the value of x+24−−−−−√−x√x+24−x?

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

Answer is E. We get different values when you test x = 1 we get 5 - 1 =4
x = 25 we get 7-5 = 2

(1) x could be 1, 4,9,25,36,49..any number ,which is a perfect square. Hence , insufficient.

(2) x could be 1, 12,25... Hence, insufficient

(1) + (2)..x could be 1 giving result 5-1=4 or x could be 25, giving result 7-5=2. Hence, Insufficient.

Answer is E.

Target question: What is the value of √(x + 24) - √x?

Given: x is a positive integer

Statement 1: √x is an integer
There are several values of x that satisfy statement 1. Here are two:
Case a: x = 1 (notice that √1 is an integer). In this case, √(x + 24) - √x = √(1 + 24) - √1 = 5 - 1 = 4
Case b: x = 4 (notice that √4 is an integer). In this case, √(x + 24) - √x = √(4 + 24) - √4 = √28 - 2
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: √(x + 24) is an integer
There are several values of x that satisfy statement 2. Here are two:
Case a: x = 1 (notice that √(1 + 24) = √25 = 5, which is an integer). In this case, √(x + 24) - √x = √(1 + 24) - √1 = 5 - 1 = 4
Case b: x = 25 (notice that √(25 + 24) = √49 = 7, which is an integer). In this case, √(x + 24) - √x = √(25 + 24) - √25 = 7 - 5 = 2
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Case a: x = 1. In this case, √(x + 24) - √x = √(1 + 24) - √1 = 5 - 1 = 4
Case b: x = 25. In this case, √(x + 24) - √x = √(25 + 24) - √25 = 7 - 5 = 2
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Statement 1 - Clearly insufficient, x can be any number that is perfect square.
Statement 2- Same explanation as statement 1
Combining both statement-

Let \(\sqrt{x+24}\)=a and \(\sqrt{x}\)=b
x+24=a^2 and x=b^2
a^2-b^2=24
(a+b)(a-b)=24
Possible Solutions
1. a+b=12 and a-b=2
a=7,b=5
2. a+b=6 and a-b=4
a=5 and b=1

Insufficient.

VeritasKarishma hi!!
Do you find another way to solve this problem?
I mean I found that x could be 1 but not that x could be 25... Do you know any trick to notice that x=1 is not the only option...
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Knowing that as numbers increase in value, the difference between their perfect squares keeps increasing helps get a small range of values to try.

Also,
1^2 = 1
2^2 = 4 (Diff from above 3)
3^2 = 9 (Diff from above 5)
4^2 = 16 (Diff from above 7)
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
10^2 = 100
11^2 = 121 (Diff from above 21)
12^2 = 144 (Diff from above 23)
13^2 = 169 (Diff from above 25)

So difference between 13^2 and any other perfect square will certainly be more than 24.
Hence, we just need to put x equal to these first few values to see when it gives another perfect square when 24 is added to it.

So start with x could be 1 such that x + 24 = 25 (Both perfect squares) - great!
x could be 4 such that x+24 = 28
x could be 9 such that x+24 = 33
x could be 16 such that x+24 = 40
x could be 25 such that x+24 = 49 (Both perfect squares) - great!

Answer (E)

(1) x can be 1, the answer will be 5-1 =4
x can be 4, the answer be \(\sqrt{4+24}\)-2
So Two different values; Insufficient.

(2) Again x can be 1 the value will 5-1 = 4, but x can also be 12, the answer will be 6-\(\sqrt{12}\)

Two diffierent asnwers.

Insufficient.

Considering both the options.
Applying the same common facts that were considered for option (1) and (2) there are two different answers.

The answer is E.

If x is a positive integer, what is the value of \(\sqrt{x+24}-\sqrt{x}\)?

(1) \(\sqrt{x}\) is an integer
Can not say. x can be any square value integer
Not sufficient

(2) \(\sqrt{x+24}\) is an integer
Can not say. x can be any square value integer-24
Not sufficient

Both statements together
There are two possibilities.
x=1 or x=25
Not sufficient

E
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Answer: Option E

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This is a value kind of DS question, so we need to find a value for the given expression knowing x is a positive integer.

From statement I alone, we can conclude that x is a perfect square.

If x = 1, value of \(\sqrt{(x+24)}- \sqrt{x}\) = 4

If x = 25, value of \(\sqrt{(x+24)}- \sqrt{x}\) = 2.

Statement I alone is insufficient to find a unique value of the expression. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, \(\sqrt{(x+24)}\) is an integer. This means (x+24) is a perfect square.

We can use the same values to prove that (x+24) is a perfect square and hence the information given in statement II will also be insufficient.
Answer option B can be eliminated. Possible answer options are C or E.

Since statement II does not add anything new to the information given in statement I or further it in any way, the combination of statements is insufficient to find a unique value for the expression. Answer option C can be eliminated.

The correct answer option is E.

Hope that helps!
Aravind B T
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I have a doubt here. Can I say that the numbers could be -7 and -5 also? Or do I have to take the under root of a number to be positive only?

We are told that "x is a positive integer" so you cannot plug negative numbers. Plus, even roots from negative numbers are undefined on the GMAT.
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