[Math Revolution GMAT math practice question]
If n is a positive integer, is √17n an integer?
1) 68n is the square of an integer.
2) n/68 is the square of an integer.
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If n is a positive integer, is √17n an integer?
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Max@Math Revolution
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\[n \geqslant 1\,\,\operatorname{int} \,\,\,\left( * \right)\]Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
If n is a positive integer, is √17n an integer?
1) 68n is the square of an integer.
2) n/68 is the square of an integer.
\[\sqrt {17 \cdot n} \,\,\mathop = \limits^? \,\,\operatorname{int} \,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\boxed{\,n\,\,\mathop = \limits^? \,\,17 \cdot L,L \geqslant 1\,\,{\text{perfect}}\,\,{\text{square}}\,}\]
\[\left( 1 \right)\,\,{2^2} \cdot 17 \cdot n = {K^2},\,\,K \geqslant 1\,\,\operatorname{int} \,\,\,\, \Leftrightarrow \,\,\,\,n = 17 \cdot {M^2},M \geqslant 1\,\,\operatorname{int} \,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{Yes}}} \right\rangle \,\,\,\,\,\,\,\,\left( {L = {M^2}} \right)\]
\[\left( 2 \right)\,\,\frac{n}{{{2^2} \cdot 17}} = {J^2},\,\,J \geqslant 1\,\,\,\operatorname{int} \,\,\,\, \Leftrightarrow \,\,\,\,n = {2^2} \cdot 17 \cdot {J^2} = 17 \cdot {\left( {2J} \right)^2}\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{Yes}}} \right\rangle \,\,\,\,\,\,\,\,\left( {L = {{\left( {2J} \right)}^2}} \right)\]
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)
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Jay@ManhattanReview
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Given: n is a positive integer.Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
If n is a positive integer, is √17n an integer?
1) 68n is the square of an integer.
2) n/68 is the square of an integer.
We have to determine whether √17n an integer.
√(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
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=>
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 √17n = 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.
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 √17n = 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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