If m and n are integers greater than 1, what is the value of

Question

GMATH  practice exercise (Quant Class 16)

If  m  and  n  are integers greater than 1, what is the value of  \,{{\root m \of {{2^n}} + \root n \of {{2^m}} } \over {{2^m} + {2^n}}}\,  ?

\left( 1 \right)\,\,m = n

\left( 2 \right)\,\,m + n = m \cdot n

thyagi · Answer

Statement 1 - m= n
If we take m=n=2
We get our final answer as 0.5
If we take m=n=3
We get our final answer as 0.25.
Therefore statement 1 is not sufficient to answer the question.

Statement 2- m+n = m.n

This statement will hold true if m=n=0 or m=n=2

But it is clearly mentioned in the question that m and n are integers greater than 1... Therefore m=n=2 has to be the only case satisfying the above condition (m+n = m.n)
 Therefore statement 2 alone is sufficient and option B is the right answer.

Posted from my mobile device

KanishkM · Answer

From 1, we can realize that m and n can take any integer values greater than 1 
That will give us different values

From 2, to m + n = mn, only one integer 2 satisfies this condition 
Therefore we can calculate an unique value

B

fskilnik · Answer

m\,,n\,\, \ge 2\,\,\,{\rm{ints}}\,\,\,\,\,\left( * \right)

?\,\, = \,\,{{\root m \of {{2^n}} + \root n \of {{2^m}} } \over {{2^m} + {2^n}}}

\left( 1 \right)\,\,m = n\,\,\,\,\left\{ \matrix{
 \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2,2} \right)\,\,\,\, \Rightarrow \,\,\,? = {{2 + 2} \over {4 + 4}} = {1 \over 2} \hfill \cr 
 \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {3,3} \right)\,\,\,\, \Rightarrow \,\,\,? = {{2 + 2} \over {8 + 8}} = {1 \over 4} \hfill \cr} \right.

\left( 2 \right)\,\,m \cdot n = m + n

\Rightarrow \,\,\,\,\,m\left( {n - 1} \right) = n\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,n - 1\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{divisor}}\,\,{\rm{of}}\,\,n\,\,\,\,\,\mathop \Rightarrow \limits_{GCF\left( {n - 1,n} \right)\,\, = \,\,1}^{\left( * \right)} \,\,\,\,\,n - 1 = 1

\left\{ \matrix{
 \,n = 2 \hfill \cr 
 \,m \cdot n = m + n \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,m = 2\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,?\,\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{SUFF}}.

The correct answer is therefore (B).

We follow the notations and rationale taught in the  GMATH  method.

Regards, 
Fabio

If m and n are integers greater than 1, what is the value of

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