combres wrote:
When the positive integer n is divided by 25, the remainder is 13. What is the value of n?
(1) n < 100
(2) When n is divided by 20, the remainder is 3.
\(\left\{ \matrix{\\
n \ge 1\,\,{\mathop{\rm int}} \hfill \cr \\
n = 25Q + 13,\,\,Q \ge 0\,\,{\mathop{\rm int}} \hfill \cr} \right.\)
\(? = n\)
\(\left( 1 \right)\,\,n < 100\,\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,{\rm{Q = 0}}\,\,\,\, \Rightarrow \,\,\,\,\,n = 13 \hfill \cr \\
\,{\rm{Take}}\,\,{\rm{Q = 1}}\,\,\,\, \Rightarrow \,\,\,\,\,n = 38 \hfill \cr} \right.\)
\(\left( 2 \right)\,\,\left\{ \matrix{\\
n = 20K + 3,\,\,K \ge 0\,\,{\mathop{\rm int}} \hfill \cr \\
n = 25Q + 13,\,\,Q \ge 0\,\,{\mathop{\rm int}} \hfill \cr} \right.\,\,\,\, \Rightarrow \,\,\,25Q + 10 = n - 3\,\, = \,\,20K\)
\(\Rightarrow \,\,\,25Q + 10\,\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,20\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,Q = 2\,\,\, \Rightarrow \,\,? = n = 63 \hfill \cr \\
\,{\rm{Take}}\,\,Q = 6\,\,\, \Rightarrow \,\,? = n = 163 \hfill \cr} \right.\)
\(\left( {1 + 2} \right)\,\,\,? = n = 63\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.