If n is a positive integer, and sqrt[(45)(14)(7^n)-(15)(7^(n-1))(54)]

Question

If n is a positive integer, and  \sqrt{(45)(14)(7^n)-(15)(7^{n-1}))(54)}  is a positive integer, what is the value of n?

1) n is prime
2) n<3

Please explain, I am having trouble with the official explanation.

ENGRTOMBA2018 · Answer

Check the question again. It seems that the question has not been transcribed properly.

Can you tell what module of manhattan prep did you find this question in?

wimpytots · Answer

Sorry I didn't notice earlier that it showed  7^n-1  instead of  7^{n-1}

ENGRTOMBA2018 · Answer

That is what I wanted to solve this question. Look below for the solution:

\sqrt{(45)(14)(7^n)-(15)(7^{n-1})(54)}  =  \sqrt{2*3^2*5*7^n *  }  =  3*20*\sqrt{7^{n-1}}  =  60*\sqrt{7^{n-1}}

As we have been given that the above expression is a positive integer --->  7^{n-1}  must be a perfect square ---> n-1 = even number such that n = ODD

Per statement 1, n = prime ---> if n = 2 --> n-1=1 but  7^1  is NOT a perfect square. Discard n =2. If n = 3 or 5 or 7, you do get  7^{n-1}  = perfect square. Thus you get multiple possible values of n making statement 1 not sufficient.

Per statement 2, n<3 ---> n = 1 or 2 are the only 2 possible values (as n = positive integer).

As mentioned above, n =2 is not an acceptable value , leaving n = 1 as the only possible case. Statement 2 is thus sufficient.

B is the correct answer.

Hope this helps.

KarishmaB · Answer

The question is still incorrect. I request you to recheck it once more.

Here is why:

\sqrt{(45)(14)(7^n)-(15)(7^{n-1}))(54)}

Taking whatever we can take common:
 \sqrt{5*9*7^{n-1}  }

\sqrt{5*9*7^{n-1} * 80}

Prime factorise all terms to get
 \sqrt{3^2 * 5^2 * 2^4 * 7^{n-1}}

3*5*2^2 \sqrt{7^{n-1}}

For this to be an integer,  7^{n-1}  should be a perfect square.

1) n is prime
n can be 3 or 5 or 7 etc since 7^2, 7^4, 7^6 etc are all perfect squares. This tells us that n cannot be 2 since 7^1 is not a perfect square. It also tells us that n cannot 1 because 1 is not prime.

2) n<3
This tells us that n is either 1 or 2. But this information contradicts the information given in statement 1. Hence the question is wrong. In DS questions, the two statements can never give contradictory data.

ENGRTOMBA2018 · Answer

VeritasPrepKarishma , excellent catch! Did not think about it and went straight to solving the question.

wimpytots · Answer

This question is definitely as posted above. It is on p.176 of GMAT Advanced Quant Second Edition.
Maybe I am just not spotting the error, I have attached a photo of the original question.

ENGRTOMBA2018 · Answer

No one is doubting the question but the point at hand is whether it is representative of a GMAT like question. GMAT will never give you a DS question wherein the 2 statements will contradict each other.

What  VeritasPrepKarishma  has mentioned is that statement 1 alone tells you that the value of n can be 3 or 5 or 7... while statement 2 alone tells you that the value of n is <3. Thus, you see that statement 1 is true with a different set of values while statement 2 gives another set of value(s). GMAT will never give such statements that contradict each other.

Had statement 2 been , n < 5, then you would have a consistent set ( atleast 1 overlap  value that satisfies both statements individually)

Hope this clears the doubt.

anilisanil · Answer

2 is a prime number too. Nothing wrong with question. So the statements do not conflict each other

Edit: got the error. Jumped the gun reading only this post. Apologies

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is

the easiest and quickest way to find the answer without actually solving the

problem. Remember equal number of variables and equations ensures a solution.

If n is a positive integer, and (45)(14)(7n)−(15)(7n−1))(54)−−−−−−−−−−−−−−−−−−−−−−−−√ is a positive integer, what is the value of n?

1) n is prime
2) n<3

Please explain, I am having trouble with the official explanation.

transforming the original condition and the question by variable approach method gives us (45)(14)(7n)−(15)(7n−1))(54)−−−−−−−−−−−−−−−−−−−−−−−−√ 
=sqrt =sqrt =sqrt 
=sqrt =integer therefore n-1=even,n=odd. There is 1 variable (n)이 and we need 1 equation to solve it. Since there is 1 each in 1) and 2), D is likely the answer. 
1) n=3,5..not sufficient, because it is not unique
2) n=1 sufficient because it is unique

therefore the answer is B.

ENGRTOMBA2018 · Answer

Your actual solution is very difficult to find in the quoted post and your advertisement. Can you please reduce the size of your advertisement text? or better put it in your signatures. This way your posts will be useful for people in this forum.

Also, put the quoted post in proper formatting, so that your actual post is clearly visible.

Thanks

If n is a positive integer, and sqrt[(45)(14)(7^n)-(15)(7^(n-1))(54)]

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If n is a positive integer, and sqrt[(45)(14)(7^n)-(15)(7^(n-1))(54)]

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