AAPL wrote:e-GMAT
N is a two-digit positive number, whose tens digit is 9. if the units digit of N^p and N^q are 6 and 4 respectively. Find N.
1) P is a positive even integer.
2) Q is a positive odd integer.
$$\left\langle N \right\rangle = {\rm{units}}\,\,{\rm{digit}}\,\,{\rm{of}}\,\,N\,\,\,$$
\[\left. \begin{gathered}
N\,\, = \,\,\underline 9 \,\underline a \,\, \hfill \\
\left\langle {{N^p}} \right\rangle = 6\,\,\, \hfill \\
\left\langle {{N^q}} \right\rangle = 4 \hfill \\
\end{gathered} \right\}\,\,\,\,\,? = N\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\boxed{\,\,? = a\,\,}\]
$$\left( 1 \right)\,\,\,p \ge 2\,\,\,{\rm{even}}\,\,\,\left\{ \matrix{
\,\,\left\langle {{{92}^4}} \right\rangle = 6\,\,\,\,{\rm{and}}\,\,\,\,\,\left\langle {{{92}^2}} \right\rangle = 4\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a = 2\,\,\,\,{\rm{viable}}\,\, \hfill \cr
\,\,\left\langle {{{94}^2}} \right\rangle = 6\,\,\,\,{\rm{and}}\,\,\,\,\,\left\langle {{{94}^1}} \right\rangle = 4\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a = 4\,\,\,{\rm{viable}}\,\, \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{INSUFF}}.\,$$
\[\left( 2 \right)\,\,\,\left\{ \begin{gathered}
\,q \geqslant 1\,\,\,{\text{odd}} \hfill \\
\,\left\langle {{N^q}} \right\rangle = 4 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{inspection}}} \,\,\,\,\,a = 4\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\text{SUFF}}{\text{.}}\]
Obs.: inspection means: if
a = 0,1,2,3,5,6,7,8 or 9, when
a is put to an odd positive power, we don´t get units digit equal to 4, as needed.
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
