If d is a positive integer, is d^1/2 an integer ?

Question

If d is a positive integer, is  \sqrt{d}  an integer ?

(1)  \sqrt{9d}  is an integer

(2)  \sqrt{10d}  is not an integer.

is oa correct???????? if then please prove....

Bunuel · Answer

Yes, it is. If d is a positive integer is \sqrt{d} an integer? Note that as d is a positive integer then \sqrt{d} is either a positive integer or an irrational number. Also note that the question basically asks whether d is a perfect square. (1) \sqrt{9d} is an integer --> \sqrt{9d}=3*\sqrt{d}=integer --> \sqrt{d}={integer} (as discussed above because d is an integer \sqrt{d} can not equal to \frac{integer}{3} ). Sufficient. (2) \sqrt{10d} is not an integer --> if d=1 then the answer will be YES but if d=2 then the answer will be NO. Not sufficient. Answer: A. Similar questions: https://gmatclub.com/forum/if-n-p-q-p-a ... 01475.html https://gmatclub.com/forum/if-d-is-a-po ... 09384.html https://gmatclub.com/forum/if-x-is-a-po ... 65976.html https://gmatclub.com/forum/if-d-is-a-po ... 67950.html https://gmatclub.com/forum/if-z-is-a-po ... 01464.html https://gmatclub.com/forum/if-y-is-a-po ... 08287.html https://gmatclub.com/forum/if-sqrt-4a-i ... 06886.html https://gmatclub.com/forum/what-is-the- ... 07195.html https://gmatclub.com/forum/is-s-an-odd- ... 06562.html

AmrithS · Answer

Statement 1:

sqrt{9d}  is an integer

=>  sqrt{3^2d}  is an integer
=> d is a perfect square
=>  sqrt{d}  is an integer.

Sufficient

Statement 2:
 sqrt{10d}  is not an integer.
=>  sqrt{2 x 5 x d}  is not an integer.
=> since 2 and 5 are not perfect squares, d may or maynot be a perfect square.

Thus  sqrt{d}  may or maynot be an integer.
e.g. 
if d = 9 then  sqrt{2 x 5 x 9}  is not an integer but  sqrt{d} is an integer
if d = 2 then  sqrt{2 x 5 x 2}  is not an integer and  sqrt{2} is not an integer
Insufficient

Ans: 'A'

144144 · Answer

clearly a. 10 doesn't have a nice square...

gmatpapa · Answer

Original answer is correct.

A.  \sqrt{9d}  is an integer: tells us 3  sqrt d  is an integer. therefore  sqrt d has to be an integer. If it is not, we will never get an integer value. Sufficient.

B.  \sqrt{10d}  is not an integer:

Case 1: Assume D to be 4
Case 2: Assume D to be 6

In both cases,  sqrt (10D)  will not be an integer (satisfies the condition)

Now in case 1,  sqrt D  is an integer but in case 2,  sqrt D  is not an integer. 2 different answers satisfy the condition. Not sufficient.

Answer A.

Juggurnaut · Answer

Why cant \sqrt{d} not be an irrational number ??

KarishmaB · Answer

Because you are given that  \sqrt{9d} = 3\sqrt{d}  is an integer. Is it possible that  3\sqrt{d}  is an integer even though  \sqrt{d}  is an irrational number? 
3*irrational number will still be an irrational number. Hence  \sqrt{d}  cannot be irrational.

siglon · Answer

The DS statements on the GMAT have some special characteristics. This question suits as a great example on how to use these to your advantage.

One of these characteristics is:  The statements never lie! 
In other words:  The statements cannot contradict each other!

On this particular question we find out that Statement 1 is sufficient, giving us the definite answer: "Yes,  \sqrt{d}  is an integer."

When moving onto Statement 2, remember that statements cannot contradict each other! We therefore know that Statement 2 will in some way give us the possibility of  \sqrt{d}  being an integer. So we do not need to check this explicitly for Statement 2.
We can move straight to searching for the possibility of  \sqrt{d}  NOT being an integer.
If we find ONE way that  \sqrt{d}  is not an integer, we know that the statement is insufficient!

Anvesh99 · Answer

Hi  KarishmaB  / Everyone

Please help me with my approach to this question.

Let us say that d=2

1) √9d is an integer :

√9*2 will this result in an integer........answer is no
can √9*4 be an integer..... answer is yes ( here I have assumed d=4)

2) √10d is not an integer.

when d =2

The statement is true √10*2 is not an integer. The same goes with any value of d

So when the question is asking us Y/N and when we know that √10d is not an integer; This could also mean that d is not an integer, because the statement 2) is correct by itself.

Let me know your thoughts and how else I should be looking at this question.

KarishmaB · Answer

Note that d cannot be 2 here because √9d is an integer but √18 is not an integer. 
√9d is same as 3 * √d and if this must be an integer then √d must be an integer.

Here d can be 2 or 4 or many other values. √20 is not an integer. √40 is not an integer etc.
but √d is not an integer when d = 2 but it is an integer when d = 4. 
Hence this statement alone is not sufficient.

Answer (A)

Anvesh99 · Answer

Thanks KarishmaB

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If d is a positive integer, is d^1/2 an integer ?

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