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For an integer n greater than 1, the set S[size=7]n[/size]={

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by Max@Math Revolution » Mon Sep 24, 2018 4:12 am

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[Math Revolution GMAT math practice question]

For an integer n greater than 1, the set Sn={ kn | k is an integer greater than or equal to 1} is defined. What is the value of n?

1) 2 is contained in the set Sn.
2) 4 is contained in the set Sn.
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For an integer n greater than 1, the set S[size=7]n[/size]={

by fskilnik@GMATH » Mon Sep 24, 2018 4:37 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

For an integer n greater than 1, the set Sn={ kn | k is an integer greater than or equal to 1} is defined. What is the value of n?

1) 2 belongs to the set Sn.
2) 4 belongs to the set Sn.
Nice problem!
\[n \geqslant 2\,\,\,\operatorname{int} \]
\[{S_n} = \left\{ {\,k \cdot n\,\,\left. {} \right|\,\,\,k \geqslant 1\,\,\operatorname{int} \,} \right\}\]
\[? = n\]
Let´s understand what each S_n means:
\[{S_2} = \left\{ {\,2k\,\,\left. {} \right|\,\,\,k \geqslant 1\,\,\operatorname{int} } \right\} = \left\{ {2,4,6,8, \ldots } \right\}\,\,{\text{ = }}\,\,{\text{positive}}\,\,{\text{even}}\,\,{\text{integers}}\]
\[{S_3} = \left\{ {\,3k\,\,\left. {} \right|\,\,\,k \geqslant 1\,\operatorname{int} } \right\} = \left\{ {3,6,9,12, \ldots } \right\}\,\,{\text{ = }}\,\,{\text{positive}}\,\,{\text{multiples}}\,\,{\text{of}}\,\,3\]
\[{S_4} = \left\{ {\,4k\,\left. {} \right|\,\,\,k \geqslant 1\,\operatorname{int} } \right\} = \left\{ {4,8,12,16, \ldots } \right\}\,\,{\text{ = }}\,\,{\text{positive}}\,\,{\text{multiples}}\,\,{\text{of}}\,\,4\]
\[\left( 1 \right)\,\,\,2 \in {S_n}\,\,\, \Rightarrow \,\,\,\,n = 2\,\, \Rightarrow \,\,\,\,{\text{SUFF}}.\]
\[\left( 2 \right)\,\,\,4 \in {S_n}\,\,\, \Rightarrow \,\,\,\,n = 2\,\,\,{\text{or}}\,\,\,n = 4\,\,\,\, \Rightarrow \,\,\,\,{\text{INSUFF}}.\]

The right answer is therefore [spoiler]__(A)_____[/spoiler] .


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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by Max@Math Revolution » Wed Sep 26, 2018 3:50 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.

Condition 1)
Only S2 = { 2, 4, 6, ... } includes 2.
Therefore, n = 2.
Condition 1) is sufficient.

Condition 2)
Two sets,
S2 = { 2, 4, 6, ... }
S4 = { 4, 8, 12, ... }
contain 4. So, n could be 2 or 4.
Since we don't have a unique solution, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

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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