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Math Expert V
Joined: 02 Sep 2009
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If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 59% (01:43) correct 41% (01:52) wrong based on 1634 sessions

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

(1) $$\sqrt{x}$$ is an integer
(2) $$\sqrt{x+24}$$ is an integer

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Target Test Prep Representative V
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Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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Bunuel wrote:
If x is a positive integer, what is the value of $$\sqrt{x+24}-\sqrt{x}$$?

(1) $$\sqrt{x}$$ is an integer
(2) $$\sqrt{x+24}$$ is an integer

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.

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Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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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
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Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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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.

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Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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E
getting possibilities more than 1

4 and 2.
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Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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Bunuel wrote:
If x is a positive integer, what is the value of $$\sqrt{x+24}-\sqrt{x}$$?

(1) $$\sqrt{x}$$ is an integer
(2) $$\sqrt{x+24}$$ is an integer

Bunuel, are these questions from OG 2017?
Math Expert V
Joined: 02 Sep 2009
Posts: 59588
Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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hsbinfy wrote:
Bunuel wrote:
If x is a positive integer, what is the value of $$\sqrt{x+24}-\sqrt{x}$$?

(1) $$\sqrt{x}$$ is an integer
(2) $$\sqrt{x+24}$$ is an integer

Bunuel, are these questions from OG 2017?

________________________
Yes.
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Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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Bunuel wrote:
hsbinfy wrote:
Bunuel wrote:
If x is a positive integer, what is the value of $$\sqrt{x+24}-\sqrt{x}$$?

(1) $$\sqrt{x}$$ is an integer
(2) $$\sqrt{x+24}$$ is an integer

Bunuel, are these questions from OG 2017?

________________________
Yes.

appreciate ur effort in putting these up.Tnks
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Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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Bunuel wrote:
If x is a positive integer, what is the value of $$\sqrt{x+24}-\sqrt{x}$$?

(1) $$\sqrt{x}$$ is an integer
(2) $$\sqrt{x+24}$$ is an integer

Statement 1- x is a square of an integer.
There are many numbers that can fulfill this condition. Not sufficient

Statement 2- x+24 is a square

x can be 1 to make the expression 25 (square of 5)

x can be 24 to make the expression 49 (square of 7)
Not sufficient

Combining two statements does not give us a unique result. Hence, E is the answer
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Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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1
Top Contributor
2
Bunuel wrote:
If x is a positive integer, what is the value of $$\sqrt{x+24}-\sqrt{x}$$?

(1) $$\sqrt{x}$$ is an integer
(2) $$\sqrt{x+24}$$ is an integer

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

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Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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1
Would this be considered a classic "C-Trap?"

How often are C-traps not actually C traps?
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Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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Would this be considered a classic "C-Trap?"

How often are C-traps not actually C traps?

1 - No, we need a finite value. Both the statements just say that the values are integers.
2 - Seldom. If are doing well on exam and then sense a c-trap then it is possibly a c-trap.

If you can correctly sense a c-trap when you are running out of time or if you have faced a difficult question, make sure not to mark C (as it's a C-trap), D (as each statement alone isn't sufficient), or E (Because it is impossible! Combining both the statements would definitely get you an answer). So you have cut down 3 choices (C, D, and E) and now there is a 50-50 chance to guess the right answer. Either A or B.

Also, if any of A or B alone seems impossible, you can pick the other answer choice and it would be correct every single time. Make sure all of this only happens only if you correctly sense a c-trap, but I hope such situation never occur to you. All the best!
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Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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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.

Bunuel wrote:
If x is a positive integer, what is the value of $$\sqrt{x+24}-\sqrt{x}$$?

(1) $$\sqrt{x}$$ is an integer
(2) $$\sqrt{x+24}$$ is an integer
Intern  Joined: 18 Aug 2018
Posts: 1
Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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Bunuel wrote:
If x is a positive integer, what is the value of $$\sqrt{x+24}-\sqrt{x}$$?

(1) $$\sqrt{x}$$ is an integer
(2) $$\sqrt{x+24}$$ is an integer

Bunuel - are the only two possibilities here x=1 and x=25? I am trying to think if there are more possibilities. My thought is that you would not be able to find more possibilities as numbers get bigger because it would be unlikely that you would be able to find values for x that could suffice x being a perfect square and x+24 being a perfect square.
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Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x  [#permalink]

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waltl wrote:
Bunuel wrote:
If x is a positive integer, what is the value of $$\sqrt{x+24}-\sqrt{x}$$?

(1) $$\sqrt{x}$$ is an integer
(2) $$\sqrt{x+24}$$ is an integer

Bunuel - are the only two possibilities here x=1 and x=25? I am trying to think if there are more possibilities. My thought is that you would not be able to find more possibilities as numbers get bigger because it would be unlikely that you would be able to find values for x that could suffice x being a perfect square and x+24 being a perfect square.

Bunuel, can we get an answer on this? In the time pressure, I struggled to see the two contradictory examples of 1 and 5 and put C. Any insight into an alternative thought process besides seeing 1 and 5 as examples? Re: If x is a positive integer, what is the value of (x + 24)^(1/2) - x   [#permalink] 07 Jul 2019, 15:12
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