If n is a positive integer, is âˆš17n an integer?

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Wed Nov 28, 2018 12:18 am

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[Math Revolution GMAT math practice question]

If n is a positive integer, is âˆš17n an integer?

1) 68n is the square of an integer.
2) n/68 is the square of an integer.

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Source: Beat The GMAT — Data Sufficiency | Wed Nov 28, 2018 12:18 am

fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Wed Nov 28, 2018 4:09 pm

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If n is a positive integer, is âˆš17n an integer?

1) 68n is the square of an integer.
2) n/68 is the square of an integer.

\[n \geqslant 1\,\,\operatorname{int} \,\,\,\left( * \right)\]
\[\sqrt {17 \cdot n} \,\,\mathop = \limits^? \,\,\operatorname{int} \,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\boxed{\,n\,\,\mathop = \limits^? \,\,17 \cdot L,L \geqslant 1\,\,{\text{perfect}}\,\,{\text{square}}\,}\]

\[\left( 1 \right)\,\,{2^2} \cdot 17 \cdot n = {K^2},\,\,K \geqslant 1\,\,\operatorname{int} \,\,\,\, \Leftrightarrow \,\,\,\,n = 17 \cdot {M^2},M \geqslant 1\,\,\operatorname{int} \,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{Yes}}} \right\rangle \,\,\,\,\,\,\,\,\left( {L = {M^2}} \right)\]

\[\left( 2 \right)\,\,\frac{n}{{{2^2} \cdot 17}} = {J^2},\,\,J \geqslant 1\,\,\,\operatorname{int} \,\,\,\, \Leftrightarrow \,\,\,\,n = {2^2} \cdot 17 \cdot {J^2} = 17 \cdot {\left( {2J} \right)^2}\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{Yes}}} \right\rangle \,\,\,\,\,\,\,\,\left( {L = {{\left( {2J} \right)}^2}} \right)\]

The correct answer is therefore (D).

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Wed Nov 28, 2018 11:09 pm

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If n is a positive integer, is âˆš17n an integer?

1) 68n is the square of an integer.
2) n/68 is the square of an integer.

Given: n is a positive integer.

We have to determine whether âˆš17n an integer.

âˆš(17n) would be an integer is n = 17m^2, where m is a positive integer

Question: Is n = 17m^2, where m is a positive integer?

Let's take each statement one by one.

1) 68n is the square of an integer.

=> 68n is a perfect square
68n = 2^2*17n
Thus, 17n is a perfect square

For 17n to be a perfect square, n = 17m^2, where m is a positive integer.

Sufficient.

2) n/68 is the square of an integer.

=> n/68 is a perfect square
n/68 = n/(2^2*17)
Thus, n/17 is a perfect square

For n/17 to be a perfect square, n = 17m^2, where m is a positive integer.

Sufficient.

The correct answer: D

Hope this helps!

-Jay
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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Fri Nov 30, 2018 12:43 am

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Modifying the question:
The question asks if âˆš17n = a for some integer a. This is equivalent to asking if 17n = a^2 for some integer a.

Since we have 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Condition 1)
Since 68n is the square of an integer and 68 = 4*17, we must have 68n = 4*17*17*k^2 for some integer k, and n = 17*k^2 or 17n = 17^2*k^2 = (17*k)^2.
Thus, 17n is the square of the integer 17k, and condition 1) is sufficient.

Condition 2)
Since n/68 is a square of an integer and 68 = 4*17, we have n/68 = m^2 for some integer m, and n = 17*4*m^2 or 17n = 17^2*2^2*m^2 = (34m)^2.
Thus, 17n is the square of the integer 17k, and condition 2) is sufficient.

Therefore, D is the answer.
Answer: D

If the original condition includes "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations" etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

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