Is (7x)^1/2 an integer?

Question

Is  \sqrt{7x}  an integer?

(1)  \sqrt{\frac{x}{7}}  is an integer

(2)  \sqrt{28x}  is an integer

Can someone please explain why point 2 is not correct? What I did is that 28 is factored into => 2*2*7 so therefore I concluded x must have at least one 7. so if x has one seven then \sqrt{7x} must be an integer. Am I doing something wrong here?

Thanks!
Alex

Thanks!
Alex

Bunuel · Accepted Answer

Is  \sqrt{7x}  an integer?

Notice that we are not told that x is an integer.

(1)  \sqrt{\frac{x}{7}}  is an integer. Given that  \sqrt{\frac{x}{7}}=integer  --> square it:  \frac{x}{7}=integer^2  -->  x=7*integer^2 . So,  \sqrt{7x}=\sqrt{7*(7*integer^2)}=7*integer=integer . Sufficient.

(2)  \sqrt{28x}  is an integer. If  x=\frac{1}{28} , then  \sqrt{7x}=\frac{1}{2}
eq{integer}  BUT if  x=0 , then  \sqrt{7x}=0=integer . Not sufficient.

Answer: A.

mau5 · Answer

From F.S 1 , we know that  \sqrt{\frac{x}{7}}  = Integer(I)-->  \sqrt{\frac{x*7}{7*7}}  =  \sqrt{7x}*\frac{1}{7}  = I-->  \sqrt{7x}  = 7*I. Thus, an integer. Sufficient.

From F.S 2, we know that  \sqrt{28x}  = Integer(I) -->  \sqrt{7*4x}  = 2* \sqrt{7x}  = I.
Thus,  \sqrt{7x}  = \frac{I}{2} . Depending on I being odd/even, the given expression will be a non-integer/integer. Insufficient.

A.

alex1233 · Answer

Thanks sorry for asking again... But could you please explain if my method/ logic was completely wrong? ie if we break out 28 int primes we see that it has 2,2,7 so x must have at least one 7? therefore we can assume that \sqrt{7x} should be integer?

Thanks again!
Alex

Bunuel · Answer

If x is NOT an integer you cannot make its prime factorization.

If we were told that x IS an integer, then  \sqrt{28x}=2\sqrt{7x}=integer . Thus x must be of the form  7^{odd}*integer^2 , so in this case  \sqrt{7x}=\sqrt{7*(7^{odd}*integer^2)}=\sqrt{7^{even}*integer^2}=integer .

Hope it's clear.

gmacforjyoab · Answer

( 1)  \sqrt{\frac{x}{7}}  is an integer  ---- lets say b
x/7 = b^2 --> where b is an integer
x= 7 *b^2
7x = (7*b) ^2
hence 7x=int^2--- sufficient

(2)  \sqrt{28x}  is an integer  ---- lets say c

28x= c^2
7x=c^2/4 = (c/2)^2
If C is odd , then ans is No and if C is even then yes . ( If you knew that x was an integer , then this would have sufficed)
Insufficient.

gmatprav · Answer

(1)  \sqrt{\frac{x}{7}}  is integer.

Let this integer be p. then

\sqrt{\frac{x}{7}} = p

\frac{x}{7} = p^2

x=7*p^2

7x=7^2*p^2

\sqrt{7x}=7*p  an integer. (1) is  sufficient

(2)  \sqrt{28x}  is integer.

Let this integer be q.

\sqrt{28x} = q

\sqrt{4*7x} = q

2*\sqrt{7x} = q

\sqrt{7x} = \frac{q}{2}

It is given that q is integer, but  \frac{q}{2}  may or may not be an integer.  not sufficient .

A is the soln.

EMPOWERgmatRichC · Answer

Hi All,

This question can be solved by TESTing VALUES, but you have to really be thorough with your TESTs (and think about what X COULD be given the information in the Facts).

We're asked if Root(7X) is an integer. This is a YES/NO question.

Fact 1: Root(X/7) is an integer.

This Fact tells us that X has to be a specific type of multiple of 7....

IF....
X = 7
the Root(7/7) = 1 and is an integer
Root(49) = 7 and the answer to the question is YES

IF....
X = 28
the Root(28/7) = 2 and is an integer
Root(196) = 14 and the answer to the question is YES

This hints at the pattern that X will be a "perfect square times a multiple of 7', which means that the answer to the question is ALWAYS YES. You can also use prime-factorization to prove that this is the case.
Fact 1 is SUFFICIENT

Fact 2: Root(28X) is an integer.

This Fact tells us that X has to be a specific type of multiple of 7....

IF....
X = 7
the Root(196) = 14 and is an integer
Root(49) = 7 and the answer to the question is YES

IF....
X = 1/28
the Root(1) = 1 and is an integer
Root(7/28) = 1/2 and the answer to the question is NO
Fact 2 is INSUFFICIENT

Final Answer:  A

GMAT assassins aren't born, they're made,
Rich

chetan2u · Answer

Hi 
lets see the what info Q has..
Is  \sqrt{(7x)}  an integer?..
possible -
a) if x ia an integer and of type y^2*7..
b) if x is a fraction of type y^2/7..

lets see the statements
(1)  \sqrt{(x/7)}  is an integer.
this tells us that x = y^2*7, where y is an integer, SAME as case (a) above
Suff

(2)  \sqrt{(28x)} is an integer.
this tells us that  \sqrt{(4*7*x)} is an integer. 
so x = y^2/28..
if y is multiple of 2, ans is YES
if y is not a multiple of 2, ans is No
Insuff

A

BrentGMATPrepNow · Answer

Target question :    Is  \sqrt{7x}  an integer?

Statement 1 :  \sqrt{\frac{x}{7}}  is an integer  
Let's say that √(x/7) = k, where k is an integer
Square both sides to get: x/7 = k²
Multiply both side by 7 to get: x = 7k²

This means 7x = 7(7k²) = 49k²
So, √(7x) = √(49k²) = 7k
Since k is an integer, we know that 7k is an integer. 
So, the answer to the target question is  YES, √(7x) IS an integer 
Since we can answer the  target question  with certainty, statement 1 is SUFFICIENT

Statement 2 :  \sqrt{28x}  is an integer 
There are several values of x that satisfy statement 2. Here are two: 
 Case a : x = 28, which means, √(7x) = √(196) = 14. So, in this case , the answer to the target question is  YES, √(7x) IS an integer 
 Case b : x = 9/28, which means, √(7x) = √(63/28) = some non-integer. So, in this case , the answer to the target question is  NO, √(7x) is NOT an integer 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

anoopdev · Answer

For statement 1, x=7 and x=343 both satisfy the condition, so how can the answer be A?

For statement 2, x= 1/28 and x=7 satisfy the condition,

But when both combined, x=1/28 is not valid. and x=7 is the unique answer.

So Answer is C, how is the answer A, can anyone guide how my method is incorrect?

EMPOWERgmatRichC · Answer

Hi anoopdev,

When dealing with DS questions, you have to make sure that you are answering the question that is ASKED (and define whether the answer is consistent - meaning that it stays the same OR that it is inconsistent - meaning that it changes).

Based on the information in Fact 1, you listed two possible values for X. Both of those values lead to the SAME answer to the question (re: YES, you would end up with an integer). That is a consistent result. All of the other potential values for X that 'fit' the information in Fact 1 will lead to the SAME answer to the given question, so Fact 1 is SUFFICIENT.

GMAT assassins aren't born, they're made,
Rich

Pritishd · Answer

Bunuel ,

Can you tell me the mistake I am making?

In A,
i. Let  x=7  then  \sqrt{\frac{x}{7}}  will be an integer and  \sqrt{7x}   =   \sqrt{7*7}   =   \sqrt{7^2}  will be an integer
ii. Let  x=21  then  \sqrt{\frac{x}{7}}  will be an integer but  \sqrt{7x}   =   \sqrt{7*7*3}  will not be an integer

Doesn't this make A insufficient?

Bunuel · Answer

If x = 21, then  \sqrt{\frac{x}{7}}=\sqrt{\frac{21}{7}}=\sqrt{3} , which is not an integer.

Pritishd · Answer

Thanks Bunuel! That was a very silly mistake from my end

Pritishd · Answer

Sharing my approach to this problem,

Statement-1   \sqrt{\frac{x}{7}}  = Int
Now let's try to modify  \sqrt{7x}  in order to get  \sqrt{\frac{x}{7}} 
 \sqrt{7x}  =  \sqrt{7x * \frac{7}{7}} 
 \sqrt{\frac{x}{7}}  *  \sqrt{49} 
We are give that  \sqrt{\frac{x}{7}}  is an integer and  \sqrt{49}  is also an integer hence  \sqrt{7x}  is an integer  ( Sufficient )

Statement-2   \sqrt{28x}  = Int
Now let's try to modify  \sqrt{7x}  in order to get  \sqrt{28x} 
 \sqrt{7x}  =  \sqrt{7x * \frac{4}{4}} 
 \sqrt{28x}  *  \sqrt{\frac{1}{4}} 
We are give that  \sqrt{28x}  is an integer (let's call it  k ) and  \sqrt{\frac{1}{4}}  yields  \frac{1}{2}

Now, for  k  *  \frac{1}{2}

If  k   =   even  then the expression is an integer and if  k   =   odd  the expression will not be an integer  ( Not Sufficient )

Ans.  A

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Is (7x)^1/2 an integer?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

Is (7x)^1/2 an integer?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards