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

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Updated on: 05 Jun 2013, 02:49
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31% (01:45) correct 69% (01:39) wrong based on 1386 sessions

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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

Originally posted by alex1233 on 13 Jan 2013, 07:22.
Last edited by Bunuel on 05 Jun 2013, 02:49, edited 3 times in total.
Renamed the topic and edited the question.
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Re: Is (7x)^1/2 an integer?  [#permalink]

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13 Jan 2013, 07:58
21
16
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 $$\sqrt{7x}=\frac{1}{2}\neq{integer}$$ BUT if $$x=0$$, then $$\sqrt{7x}=0=integer$$. Not sufficient.

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Re: Is (7x)^1/2 an integer?  [#permalink]

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05 Jun 2013, 22:53
5
alex1233 wrote:
Is $$\sqrt{7x}$$ an integer?

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

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

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.
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Re: Is (7x)^1/2 an integer?  [#permalink]

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13 Jan 2013, 08:01
[quote="alexpavlos"]Is $$\sqrt{7x}$$ an integer?

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

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

for sqt 7x to be an integer , x has to have the form 7^a*z^b, where a is an odd integer and z is an integer and b a an even intiger.

from 1

x is in the form 7^a*z^b

from 2
2sqt 7x is an integer....this means that sqt 7x is in the form k/2 however , it says nothing about the form of x itself... thus insuff

A
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Re: Is (7x)^1/2 an integer?  [#permalink]

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13 Jan 2013, 08:16
1
Bunuel wrote:
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 $$\sqrt{7x}=\frac{1}{2}\neq{integer}$$ BUT if $$x=0$$, then $$\sqrt{7x}=0=integer$$. Not sufficient.

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
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Re: Is (7x)^1/2 an integer?  [#permalink]

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13 Jan 2013, 08:29
2
alexpavlos wrote:
Bunuel wrote:
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 $$\sqrt{7x}=\frac{1}{2}\neq{integer}$$ BUT if $$x=0$$, then $$\sqrt{7x}=0=integer$$. Not sufficient.

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

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.
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Re: Is (7x)^1/2 an integer?  [#permalink]

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26 May 2013, 20:17
2
alex1233 wrote:
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

Quote:
$$\sqrt{7x}$$ = integer ? or 7x= int^2 ?

(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.
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Re: Is (7x)^1/2 an integer?  [#permalink]

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05 Jun 2013, 02:55
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Re: Is (7x)^1/2 an integer?  [#permalink]

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21 Dec 2013, 05:09
(1) $$\sqrt{7x}$$ = 7$$\sqrt{\frac{x}{7}}$$

7 and $$\sqrt{\frac{x}{7}}$$ are integers so $$\sqrt{7x}$$ is an integer---Sufficient

(2) $$\sqrt{7x}$$ = $$\sqrt{28x}/2$$

We know that $$\sqrt{28x}$$ is an integer but not sure if it is divisible by 2 or not---Insufficient

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Re: Is (7x)^1/2 an integer?  [#permalink]

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03 Feb 2014, 05:29
2
1
(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.
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Re: Is (7x)^1/2 an integer?  [#permalink]

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15 Apr 2014, 07:04
(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.

tnx for this
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Re: Is (7x)^1/2 an integer?  [#permalink]

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15 Apr 2014, 07:09
gmatonline wrote:
(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.

tnx for this

Hope it helps.
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Re: Is (7x)^1/2 an integer?  [#permalink]

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16 Apr 2015, 10:54
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

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Special Offer: Save $75 + GMAT Club Tests Free Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/ *****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***** Intern Joined: 24 Nov 2015 Posts: 5 Location: India GMAT 1: 680 Q49 V34 Is [m][square_root](7x)[/square_root][/m] an integer? [#permalink] ### Show Tags 15 May 2016, 08:36 Is $$\sqrt{(7x)}$$ an integer? (1) $$\sqrt{(x/7)}$$ is an integer. (2) $$\sqrt{(28x)}$$is an integer. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. EITHER statement BY ITSELF is sufficient to answer the question. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, requiring more data pertaining to the problem. Math Expert Joined: 02 Sep 2009 Posts: 52907 Re: Is (7x)^1/2 an integer? [#permalink] ### Show Tags 15 May 2016, 08:41 sahilhanda wrote: Is $$\sqrt{(7x)}$$ an integer? (1) $$\sqrt{(x/7)}$$ is an integer. (2) $$\sqrt{(28x)}$$is an integer. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. EITHER statement BY ITSELF is sufficient to answer the question. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, requiring more data pertaining to the problem. Merging topics. Please refer to the discussion above. _________________ Math Expert Joined: 02 Aug 2009 Posts: 7334 Re: Is [m][square_root](7x)[/square_root][/m] an integer? [#permalink] ### Show Tags 15 May 2016, 08:45 sahilhanda wrote: Is $$\sqrt{(7x)}$$ an integer? (1) $$\sqrt{(x/7)}$$ is an integer. (2) $$\sqrt{(28x)}$$is an integer. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. EITHER statement BY ITSELF is sufficient to answer the question. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, requiring more data pertaining to the problem. 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 _________________ 1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html 4) Base while finding % increase and % decrease : https://gmatclub.com/forum/percentage-increase-decrease-what-should-be-the-denominator-287528.html GMAT Expert Intern Joined: 18 Sep 2015 Posts: 22 Re: Is (7x)^1/2 an integer? [#permalink] ### Show Tags 16 May 2016, 22:51 alex1233 wrote: 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 A: let's say $$\sqrt{\frac{x}{7}}$$ = m $$\sqrt{\frac{x*7}{7*7}}$$ = m $$\sqrt{7x}$$/7 = m => $$\sqrt{7x}$$ = 7*m => $$\sqrt{7x}$$ is an integer. (m is given to be an integer) A is sufficient. B: let's say $$\sqrt{28x}$$ = k 2*$$\sqrt{7x}$$ = k $$\sqrt{7x}$$ = k/2 Here the answer depends on the parity (evenness or oddness) of k and parity of k is unknown. Hence B is not sufficient. Final answer: A CEO Joined: 11 Sep 2015 Posts: 3431 Location: Canada Re: Is (7x)^1/2 an integer? [#permalink] ### Show Tags 14 Oct 2018, 15:34 Top Contributor alex1233 wrote: Is $$\sqrt{7x}$$ an integer? (1) $$\sqrt{\frac{x}{7}}$$ is an integer (2) $$\sqrt{28x}$$ is an integer 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 _________________ Test confidently with gmatprepnow.com Intern Joined: 02 Jan 2019 Posts: 1 Re: Is (7x)^1/2 an integer? [#permalink] ### Show Tags 28 Jan 2019, 07:29 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? EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 13546 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Is (7x)^1/2 an integer? [#permalink] ### Show Tags 28 Jan 2019, 11:38 anoopdev wrote: 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? 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. 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Re: Is (7x)^1/2 an integer?   [#permalink] 28 Jan 2019, 11:38
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