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

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Rospino: Newbie | Next Rank: 10 Posts; Posts: 8; Joined: Mon May 07, 2012 12:12 pm

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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Source: Beat The GMAT — Data Sufficiency | Mon Mar 31, 2014 4:47 pm

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Mar 31, 2014 4:53 pm

Rospino wrote:If x is a positive integer, is âˆšx an integer?
(1) âˆš(4x) is an integer
(2) âˆš(3x) is not an integer

Target question: Is âˆšx an integer?

Given: x is a positive integer

Statement 1: âˆš(4x) is an integer
IMPORTANT CONCEPT: If K is an integer, then âˆšK will be an integer if the prime factorization of K has an even number of each prime.
Some examples:
âˆš144 = 12 (integer), and 144 = (2)(2)(2)(2)(3)(3) [four 2's and two 3's]
âˆš1600 = 40 (integer), and 1600 = (2)(2)(2)(2)(2)(2)(5)(5) [six 2's and two 5's]
âˆš441 = 21 (integer), and 441 = (3)(3)(7)(7)[two 3's and two 7's]
âˆš12 = some non-integer, and 12 = (2)(2)(3)[two 2's and one 3's]

So, if âˆš(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 âˆšx must be an integer.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: âˆš(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 âˆš(3x) = âˆš12, which is not an integer. In this case, âˆšx is an integer.
Case b: x = 5. This means that âˆš(3x) = âˆš15, which is not an integer. In this case, âˆšx is not an integer.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Rospino: Newbie | Next Rank: 10 Posts; Posts: 8; Joined: Mon May 07, 2012 12:12 pm

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 âˆšK will be an integer if the prime factorization of K has an even number of each prime.

Regards,
Ricardo

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

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

Brent Hanneson - Creator of GMATPrepNow.com

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Amrabdelnaby: Master | Next Rank: 500 Posts; Posts: 137; Joined: Fri Nov 13, 2015 11:01 amFollowed by: 2 members; Thanked: 1 times

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

Brent,

Question though!

Now âˆš4x is equal to 2âˆšx, so if âˆšx yields to 1.5 or 2.5 or 3.5 etc... âˆš4x or 2âˆšx will be always an integer.

so how come this statement is sufficient?

why did we assume that âˆšx must be an integer if âˆš4x is an integer although âˆš4x =2âˆšx

Brent@GMATPrepNow wrote:
Rospino wrote:If x is a positive integer, is âˆšx an integer?
(1) âˆš(4x) is an integer
(2) âˆš(3x) is not an integer

Target question: Is âˆšx an integer?

Given: x is a positive integer

Statement 1: âˆš(4x) is an integer
IMPORTANT CONCEPT: If K is an integer, then âˆšK will be an integer if the prime factorization of K has an even number of each prime.
Some examples:
âˆš144 = 12 (integer), and 144 = (2)(2)(2)(2)(3)(3) [four 2's and two 3's]
âˆš1600 = 40 (integer), and 1600 = (2)(2)(2)(2)(2)(2)(5)(5) [six 2's and two 5's]
âˆš441 = 21 (integer), and 441 = (3)(3)(7)(7)[two 3's and two 7's]
âˆš12 = some non-integer, and 12 = (2)(2)(3)[two 2's and one 3's]

So, if âˆš(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 âˆšx must be an integer.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: âˆš(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 âˆš(3x) = âˆš12, which is not an integer. In this case, âˆšx is an integer.
Case b: x = 5. This means that âˆš(3x) = âˆš15, which is not an integer. In this case, âˆšx 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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theCEO: Master | Next Rank: 500 Posts; Posts: 363; Joined: Sun Oct 17, 2010 3:24 pmFollowed by: 3 members; Thanked: 115 times

by theCEO » Sun Nov 22, 2015 2:31 pm

Now âˆš4x is equal to 2âˆšx, so if âˆšx yields to 1.5 or 2.5 or 3.5 etc... âˆš4x or 2âˆšx will be always an integer.

so how come this statement is sufficient?

why did we assume that âˆšx must be an integer if âˆš4x is an integer although âˆš4x =2âˆšx

Brent@GMATPrepNow wrote:
Rospino wrote:If x is a positive integer, is âˆšx an integer?
(1) âˆš(4x) is an integer
(2) âˆš(3x) is not an integer

Hi Amrabdelnaby,

I am not Brent

but let's adress this statement: "Now âˆš4x is equal to 2âˆšx, so if âˆšx yields to 1.5 or 2.5 or 3.5 etc... âˆš4x or 2âˆšx will be always an integer."

If âˆšx = 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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Amrabdelnaby: Master | Next Rank: 500 Posts; Posts: 137; Joined: Fri Nov 13, 2015 11:01 amFollowed by: 2 members; Thanked: 1 times

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 âˆš4x is also an integer then âˆšx 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 âˆš4x is equal to 2âˆšx, so if âˆšx yields to 1.5 or 2.5 or 3.5 etc... âˆš4x or 2âˆšx will be always an integer.

so how come this statement is sufficient?

why did we assume that âˆšx must be an integer if âˆš4x is an integer although âˆš4x =2âˆšx

Brent@GMATPrepNow wrote:
Rospino wrote:If x is a positive integer, is âˆšx an integer?
(1) âˆš(4x) is an integer
(2) âˆš(3x) is not an integer

Hi Amrabdelnaby,

I am not Brent but let's adress this statement: "Now âˆš4x is equal to 2âˆšx, so if âˆšx yields to 1.5 or 2.5 or 3.5 etc... âˆš4x or 2âˆšx will be always an integer."

If âˆšx = 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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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Sun Nov 22, 2015 2:38 pm

Amrabdelnaby wrote:Brent,

Question though!

Now âˆš4x is equal to 2âˆšx, so if âˆšx yields to 1.5 or 2.5 or 3.5 etc... âˆš4x or 2âˆšx will be always an integer.

so how come this statement is sufficient?

why did we assume that âˆšx must be an integer if âˆš4x is an integer although âˆš4x =2âˆšx

âˆšx CAN NEVER equal something.5
If âˆšx WERE to equal something.5, then x = (something.5)Â² = 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

Brent Hanneson - Creator of GMATPrepNow.com

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Fri Nov 27, 2015 1:35 am

An algebraic way of seeing this:

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

Squaring both sides, we have

x = mÂ² + m + .25

But this says x = (integer)Â² + (integer) + .25, a contradiction. Hence x and m cannot both be integers.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8139; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

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

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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