If n is a positive integer greater than 1, is (n + 11)^(1/2) = n^(1/2)

Question

If n is a positive integer greater than 1, is  \sqrt{n+11} = \sqrt{n} + 1 ?

(1) n is a perfect square
(2) n + 11 is a perfect square.

WizakoBaskar · Accepted Answer

The data is sufficient if we can determine whether the equation holds good with the information in the statements.

n is a positive integer. 
 Properties of perfect squares . The difference between squares of consecutive positive integers will increase by 2. 
For instance, 2^2 - 1^2 =  3 ; 3^2 - 2^2 =  5 ; 4^2 - 3^2 =  7 ; 5^2 - 4^2 =  9 ; 6^2 - 5^2 =  11  and so on.

Statement 1 : n is a perfect square.
If n = 4;  \sqrt{4+11} 
eq \sqrt{4} + 1 
If n = 25;  \sqrt{25+11} = \sqrt{25} + 1

Cannot determine whether the equality will hold good. Statement 1 alone is not sufficient.

Statement 2 : n + 11 is a perfect square
If n = 14; n + 11 = 25, which is a perfect square.  \sqrt{14+11} 
eq \sqrt{14} + 1 
If n = 25; n + 11 = 36, which is a perfect square.  \sqrt{25+11} = \sqrt{25} + 1

Combining the two statements : n is a perfect square and (n + 11) is also a perfect square.
If the equation were to hold good, the two numbers n and (n + 11) will have to be squares of two consecutive numbers.
From the properties of difference between square of positive integers, there can exist only one set of values where the difference between squares of consecutive numbers is 11. 
The difference between 6^2 and 5^2 is 11 and we cannot find any other set of two consecutive integers that will satisfy this condition. i.e., the only value that satisfies is when n = 25 and (n + 11) = 36.

Choice C is the answer.

Tulkin987 · Answer

Even though official answer is  E , I will go with  C .
Because,  (1)  and  (2)  taken together, results in  n=25 . This is sufficient to answer the question.

Viserion99 · Answer

Yes. 
Thanks for the reassurance.
Even, I went with C.

Sent from my Moto G (5S) Plus using  GMAT Club Forum mobile app

tejyr · Answer

\sqrt{n+11} = \sqrt{n} + 1 ?
Squaring on both ides and solve for n then we will get n=25.So we need to get a unique value 25 from A or b or both
(1) n is a perfect square -25,200,4,9,...
(2) n + 11 is a perfect square.-25,14,5,....

n=25 is only perfect square with n+11 a perfect square so unique and C is the answer.

Mo2men · Answer

Dear  Bunuel ,

I have got same answer C as the above. Can you change the OA?

jayantbakshi · Answer

Dear  Bunuel ,

Shouldn't the answer be E? I see a lot of my friends have said C, considering that n=25 and n+11 = 36, would solve the problem.

I say that because \sqrt{36} would be +/- 6 and not just =6. Similarly, the \sqrt{25}, will be +/-5 and not just +5.
Given this, I believe the ans to be E. I note that the restriction to be a positive integer is for n, not on \sqrt{n}.

Requesting your expertise! Thanks a lot.

Bunuel · Answer

When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root.   That is:

\sqrt{9} = 3 , NOT +3 or -3;
 \sqrt {16} = 2 , NOT +2 or -2;

Notice that in contrast, the equation  x^2 = 9  has TWO solutions, +3 and -3. Because  x^2 = 9  means that  x =-\sqrt{9}=-3  or  x=\sqrt{9}=3 .

jayantbakshi · Answer

When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root. That is:

\(\sqrt{9} = 3\), NOT +3 or -3;
\(\sqrt[4]{16} = 2\), NOT +2 or -2;

Notice that in contrast, the equation \(x^2 = 9\) has TWO solutions, +3 and -3. Because \(x^2 = 9\) means that \(x =-\sqrt{9}=-3\) or \(x=\sqrt{9}=3\).[/quote]

Thanks so much Bunuel, this was very helpful! Very kind of you.

Much appreciated

shubhamjha22 · Answer

Hey, Here's what I did. can anyone help me understand where did I go wrong? TIA

Question is (n+11)^0.5= (n)^0.5 +1
= n+11= n+1 + 2*(n)^0.5
= n^0.5= 5 or is n=25?

Statement 1: not sufficient
Statement 2: (n+11)= k^2, let n =a^2=> a^2 +11 = K^2=> K^2-a^2=11=> (K+a)(k-a)=> 11. Since 11 is prime 
therefore, K-a=1 and K+a = 11, only pair of positive integers are 6 & 5 hence stament 2 is Sufficient. Requesting help guys. Many thanks in advance

IanStewart · Answer

I've highlighted the issue in red: you seem to be accidentally using both Statements at once when you analyze Statement 2 alone, or at least you are when you assume  a  is an integer. Everything else about your solution is perfect though.

I know this isn't what Bunuel means to suggest, but in case anyone reading misunderstands the above: if you see a square root ('radical') symbol  anywhere , that always means "the non-negative root". Whether you're in a high school algebra class or at an international math conference or taking the GMAT, √36 is equal to 6, never to -6. That's the definition of the "√" symbol. I mention this just because there's an unfortunately widespread misconception that there is such a thing as "GMAT math" that has its own idiosyncratic rules, and that test takers need to learn the special rules and definitions that are part of "GMAT math". That's not true; GMAT math is identical to the math you learn anywhere else.

shubhamjha22 · Answer

That was a bone headed move. Many thanks Ian.

Posted from my mobile device

bond001 · Answer

@chetan2u,@Bunuel,@IanStewart

I don,t know where I am making a mistake 
Statment 1 
if n=25 equation satisfy but I am not able to get any other n that satisfy this equation then how can I say that statement 1 is not sufficient 
statement 2 
again same problem as statement 1 i know only one value of n i.e 25

Answer D

MathRevolution · Answer

Let's solve the question using the Variable Approach which says modify the question should be the first step.

On squaring both the sides, we get

=> n + 11 = n + 1 + 2  n^{\frac{1}{2}}

=> 10 = 2 n^{\frac{1}{2}}

=> 5 =  n^{\frac{1}{2}}

=> 25 = n

So, we have to find whether n = 25.

Variable Approach second and the third step says to match the number of variables from the original conditions with the number of equations available from both the conditions. We have 1 variable (n) and to match it, we need 1 equation. We will get 1 equation each from condition number (1) and condition number (2). So, the most likely answer should be D.

Let's look at each condition separately.

Condition(1) tells us that 'n' is a perfect square.

Possible values of n = 16, 25, 36 and more. - Is n = 25 = YES and NO.

Since the answer is not a YES or a NO, condition(1) alone is not sufficient according to CMT(1) which states that answer should be a unique YES or a NO.

Condition(2) tells us that 'n + 11' is a perfect square.

If n = 14, then 14 + 11 = 25 which is a perfect square - Is n = 25 - NO
If n = 25, then 25 + 11 = 36 which is a perfect square - Is n = 25 - YES

Since the answer is not a YES or a NO, condition(2) alone is not sufficient according to CMT(1) which states that answer should be a unique YES or a NO.

Combining both the equations tells us that n is a perfect square and n + 11 is also a perfect square. Hence n = 25 - YES

Since the answer is a unique YES, both conditions combined together are sufficient according to CMT(1) which states that the answer should be a unique YES or a NO.

Both conditions combined together are sufficient.

C is the answer

Answer C

justbegan · Answer

here i suggest another method to solve the problem.

let us square both sides of the equation and we get :

n+11=n+2\sqrt{n}+1
2\sqrt{n}=10
\sqrt{n}=5

condition 1 is to vague
condition 2 is again vague because 27+11=36 is a perfect square, 89+11=100 is a perfect square

condition 1+ condition 2 says n is perfect square and n+11=perfect square and this leads to n=25 and thus \sqrt{n}=5 which is sufficient.

hence C

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If n is a positive integer greater than 1, is (n + 11)^(1/2) = n^(1/2)

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If n is a positive integer greater than 1, is (n + 11)^(1/2) = n^(1/2)

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