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If x is a positive integer, is sqrt(x) an integer?

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If x is a positive integer, is sqrt(x) an integer?

by Rospino » Mon Mar 31, 2014 4:47 pm
If x is a positive integer, is sqrt(x) an integer?
(1) sqrt(4x) is an integer
(2) sqrt(3x) is not an integer

OA: A

I used the plug in number approach. However, this problem took me about 3.5 mins - 1.5 mins too long. Any quicker ways?
(1) sqrt(4x) is an integer
a. if x = 1, then sqrt(4x) = 2, then sqrt(x) = 1, which is an integer.
b. if x = 4, then sqrt(4x) = 4, then sqrt(x) = 2, which is an integer.
c. if x = 9, then sqrt(4x) = 6, then sqrt(x) = 3, which is an integer.

Sufficient. Eliminate [spoiler]BCE; AD[/spoiler]Remains.

(2) sqrt(3x) is not an integer
a. if x = 2, then sqrt(3x) = sqrt(6), then sqrt(x) = sqrt(2), which is not an integer.
b. if x = 4, then sqrt(3x) = sqrt(12), then sqrt(x) = sqrt(4), which is an integer.

Insufficient. Eliminate D
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by Brent@GMATPrepNow » Mon Mar 31, 2014 4:53 pm
Rospino wrote:If x is a positive integer, is √x an integer?
(1) √(4x) is an integer
(2) √(3x) is not an integer

Target question: Is √x an integer?

Given: x is a positive integer

Statement 1: √(4x) is an integer
IMPORTANT CONCEPT: If K is an integer, then √K will be an integer if the prime factorization of K has an even number of each prime.
Some examples:
√144 = 12 (integer), and 144 = (2)(2)(2)(2)(3)(3) [four 2's and two 3's]
√1600 = 40 (integer), and 1600 = (2)(2)(2)(2)(2)(2)(5)(5) [six 2's and two 5's]
√441 = 21 (integer), and 441 = (3)(3)(7)(7)[two 3's and two 7's]
√12 = some non-integer, and 12 = (2)(2)(3)[two 2's and one 3's]

So, if √(4x) is an integer, then the prime factorization of 4x has an even number of each prime.
Since 4x = (2)(2)(x) we can see that the prime factorization of x must have an even number of each prime.
If the prime factorization of x has an even number of each prime, then √x must be an integer.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: √(3x) is not an integer.
There are several values of x that meet this condition. Here are two:
Case a: x = 4. This means that √(3x) = √12, which is not an integer. In this case, √x is an integer.
Case b: x = 5. This means that √(3x) = √15, which is not an integer. In this case, √x is not an integer.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent
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by Rospino » Mon Mar 31, 2014 5:04 pm
Brent,

Thank you very much for the explanation. It was very clear. Also, the concept you mentioned will definitely save me time on the exam.
IMPORTANT CONCEPT: If K is an integer, then √K will be an integer if the prime factorization of K has an even number of each prime.
Regards,
Ricardo
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by Brent@GMATPrepNow » Mon Mar 31, 2014 5:05 pm
You're welcome.
It's a frequently-tested concept, so good to know.

Cheers,
Brent
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by Amrabdelnaby » Sun Nov 22, 2015 2:00 pm
Brent,

Question though!

Now √4x is equal to 2√x, so if √x yields to 1.5 or 2.5 or 3.5 etc... √4x or 2√x will be always an integer.

so how come this statement is sufficient?

why did we assume that √x must be an integer if √4x is an integer although √4x =2√x
Brent@GMATPrepNow wrote:
Rospino wrote:If x is a positive integer, is √x an integer?
(1) √(4x) is an integer
(2) √(3x) is not an integer

Target question: Is √x an integer?

Given: x is a positive integer

Statement 1: √(4x) is an integer
IMPORTANT CONCEPT: If K is an integer, then √K will be an integer if the prime factorization of K has an even number of each prime.
Some examples:
√144 = 12 (integer), and 144 = (2)(2)(2)(2)(3)(3) [four 2's and two 3's]
√1600 = 40 (integer), and 1600 = (2)(2)(2)(2)(2)(2)(5)(5) [six 2's and two 5's]
√441 = 21 (integer), and 441 = (3)(3)(7)(7)[two 3's and two 7's]
√12 = some non-integer, and 12 = (2)(2)(3)[two 2's and one 3's]

So, if √(4x) is an integer, then the prime factorization of 4x has an even number of each prime.
Since 4x = (2)(2)(x) we can see that the prime factorization of x must have an even number of each prime.
If the prime factorization of x has an even number of each prime, then √x must be an integer.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: √(3x) is not an integer.
There are several values of x that meet this condition. Here are two:
Case a: x = 4. This means that √(3x) = √12, which is not an integer. In this case, √x is an integer.
Case b: x = 5. This means that √(3x) = √15, which is not an integer. In this case, √x is not an integer.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent
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by theCEO » Sun Nov 22, 2015 2:31 pm
Now √4x is equal to 2√x, so if √x yields to 1.5 or 2.5 or 3.5 etc... √4x or 2√x will be always an integer.

so how come this statement is sufficient?

why did we assume that √x must be an integer if √4x is an integer although √4x =2√x
Brent@GMATPrepNow wrote:
Rospino wrote:If x is a positive integer, is √x an integer?
(1) √(4x) is an integer
(2) √(3x) is not an integer

Hi Amrabdelnaby,

I am not Brent :) but let's adress this statement: "Now √4x is equal to 2√x, so if √x yields to 1.5 or 2.5 or 3.5 etc... √4x or 2√x will be always an integer."

If √x = 1.5
x = (1.5)^2 = 2.25

The prompt tells us that x is a positive number: therefore x could never be 2.25 or any non-integer
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by Amrabdelnaby » Sun Nov 22, 2015 2:38 pm
Oh my!

I missed that part of the question!

what a shame :)

I totally agree with you CEO.. if x itself is an +ve integer and √4x is also an integer then √x must be an integer because x must contain an even number of prime factors in this case, since we eliminated the possibility of x being a fraction.

I see what you are saying now.

Thanks man :)
theCEO wrote:Now √4x is equal to 2√x, so if √x yields to 1.5 or 2.5 or 3.5 etc... √4x or 2√x will be always an integer.

so how come this statement is sufficient?

why did we assume that √x must be an integer if √4x is an integer although √4x =2√x
Brent@GMATPrepNow wrote:
Rospino wrote:If x is a positive integer, is √x an integer?
(1) √(4x) is an integer
(2) √(3x) is not an integer

Hi Amrabdelnaby,

I am not Brent :) but let's adress this statement: "Now √4x is equal to 2√x, so if √x yields to 1.5 or 2.5 or 3.5 etc... √4x or 2√x will be always an integer."

If √x = 1.5
x = (1.5)^2 = 2.25

The prompt tells us that x is a positive number: therefore x could never be 2.25 or any non-integer
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by Brent@GMATPrepNow » Sun Nov 22, 2015 2:38 pm
Amrabdelnaby wrote:Brent,

Question though!

Now √4x is equal to 2√x, so if √x yields to 1.5 or 2.5 or 3.5 etc... √4x or 2√x will be always an integer.

so how come this statement is sufficient?

why did we assume that √x must be an integer if √4x is an integer although √4x =2√x
√x CAN NEVER equal something.5
If √x WERE to equal something.5, then x = (something.5)² = somenumber.25
Here we have a contradiction, because the given information that says x is an INTEGER, and an integer cannot end in .25

Cheers,
Brent
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by Matt@VeritasPrep » Fri Nov 27, 2015 1:35 am
An algebraic way of seeing this:

Suppose that √x = m + .5, where x and m are integers.

Squaring both sides, we have

x = m² + m + .25

But this says x = (integer)² + (integer) + .25, a contradiction. Hence x and m cannot both be integers.
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Re: If x is a positive integer, is sqrt(x) an integer?

by Scott@TargetTestPrep » Sun Jul 04, 2021 4:21 am
Rospino wrote: ↑
Mon Mar 31, 2014 4:47 pm
 If x is a positive integer, is sqrt(x) an integer?
(1) sqrt(4x) is an integer
(2) sqrt(3x) is not an integer

OA: A

Solution:

Question Stem Analysis:

We need to determine whether √x is an integer, given that x is a positive integer. Notice that √x is an integer if x is a perfect square.

Statement One Alone:

Since √(4x) = 2√x is an integer, √x is either itself an integer or it is 1/2 of some integer. If √x is the former case, we are done. If √x is the latter case, we can let √x = k/2 where k is some positive integer. Squaring both sides, we have:

x = k^2/4

Since x is a positive integer, we see that k has to be even. If k is even, then k/2, or √x, will be an integer. Statement one alone is sufficient.

Statement Two Alone:

Statement two is not sufficient. For example, if x = 4, then √x = 2 is an integer (notice that √(3x) = √12 is not an integer). However, if x = 5, then √x = √5 is not integer (notice that √(3x) = √15 is also not an integer). Statement two alone is not sufficient.

Answer: A

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