swerve 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.
Source: GMAT Prep
$$\left\{ \matrix{
n \ge 1\,\,{\mathop{\rm int}} \hfill \cr
n = 25Q + 13,\,\,Q \ge 0\,\,{\mathop{\rm int}} \hfill \cr} \right.$$
$$? = n$$
$$\left( 1 \right)\,\,n < 100\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,{\rm{Q = 0}}\,\,\,\, \Rightarrow \,\,\,\,\,n = 13 \hfill \cr
\,{\rm{Take}}\,\,{\rm{Q = 1}}\,\,\,\, \Rightarrow \,\,\,\,\,n = 38 \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\left\{ \matrix{
n = 20K + 3,\,\,K \ge 0\,\,{\mathop{\rm int}} \hfill \cr
n = 25Q + 13,\,\,Q \ge 0\,\,{\mathop{\rm int}} \hfill \cr} \right.\,\,\,\, \Rightarrow \,\,\,25Q + 10 = n - 3\,\, = \,\,20K$$
$$ \Rightarrow \,\,\,25Q + 10\,\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,20\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,Q = 2\,\,\, \Rightarrow \,\,? = n = 63 \hfill \cr
\,{\rm{Take}}\,\,Q = 6\,\,\, \Rightarrow \,\,? = n = 163 \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,? = n = 63$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
