BTGmoderatorDC wrote:If x and y are nonnegative integers, what is the value of y?

(1) 3^x = 5^y
(2) |y| = −y
Source: Veritas Prep

\[x,y \geqslant 0\,\,\,{\text{ints}}\,\,\,\,\left( * \right)\]
\[? = y\]
\[\left( 1 \right)\,\,\,\left\{ \begin{gathered}
\,\,{3^x}\,\left[ {{\text{and}}\,\,\left( * \right)} \right]\,\,\, = \,\,\,\underline 1 \,\,\,{\text{or}}\,\,{\text{only}}\,\,3\,\,{\text{as}}\,\,{\text{prime}}\,\,{\text{factor}} \hfill \\
\,\,{5^y}\,\left[ {{\text{and}}\,\,\left( * \right)} \right]\,\,\, = \,\,\,\underline 1 \,\,\,{\text{or}}\,\,{\text{only}}\,\,5\,\,{\text{as}}\,\,{\text{prime}}\,\,{\text{factor}} \hfill \\
\end{gathered} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,\,\underline {x = y = 0} \,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.\]
\[\left( 2 \right)\,\,\,\left| y \right| = - y\,\,\,\, \Leftrightarrow \,\,\,\,\,y \leqslant 0\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,y = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.\]

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

Regards,
Fabio.