fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 16)

If m and n are integers greater than 1, what is the value of A(m,n) ?
$$A\left( {m,n} \right) = {{\root m \of {{2^n}} + \root n \of {{2^m}} } \over {{2^m} + {2^n}}}$$
$$\left( 1 \right)\,\,m = n$$
$$\left( 2 \right)\,\,m + n = m \cdot n$$

$$m\,,n\,\, \ge 2\,\,\,{\rm{ints}}\,\,\,\,\,\left( * \right)$$
$$?\,\, = \,\,A\left( {m,n} \right)\,\, = \,\,{{\root m \of {{2^n}} + \root n \of {{2^m}} } \over {{2^m} + {2^n}}}$$

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

$$\left( 2 \right)\,\,m \cdot n = m + n$$
$$ \Rightarrow \,\,\,\,\,m\left( {n - 1} \right) = n\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,n - 1\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{divisor}}\,\,{\rm{of}}\,\,n\,\,\,\,\,\mathop \Rightarrow \limits_{GCF\left( {n - 1,n} \right)\,\, = \,\,1}^{\left( * \right)} \,\,\,\,\,n - 1 = 1$$
$$\left\{ \matrix{
\,n = 2 \hfill \cr
\,m \cdot n = m + n \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,m = 2\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = A\left( {2,2} \right)\,\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{SUFF}}.$$


The correct answer is therefore (B).


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

Regards,
Fabio.