What is the remainder when a positive integer t is divided by 15?

Very beautiful problem, akurathi12 ... Congrats and kudos!


\(t \ge 1\,\,{\mathop{\rm int}} \,\,\left( * \right)\)

\(t = 15M + R\)

\(M\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,,\,\,\,0 \le R\,\,{\mathop{\rm int}} \,\, \le 14\)

\(? = R\)


\(\left( 1 \right)\,\,t = 45K + 13 = 15\left( {3K} \right) + 13\,\,,\,\,K\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{\\
\,M = 3K \hfill \cr \\
\,R = 13\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}. \hfill \cr} \right.\)


\(\left( 2 \right)\,\,\left\{ \matrix{\\
\,t = 10J + 3\,\,\,,\,\,\,J\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,3} \,\,\,\,\,\,3t = 30J + 9 \hfill \cr \\
t = 6L + 1\,\,\,,\,\,\,L\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,5} \,\,\,\,\,\,5t = 30L + 5 \hfill \cr} \right.\,\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( - \right)} \,\,\,\,\,2t = 30\left( {L - J} \right) - 4\)

\(2t = 30\left( {L - J} \right) - 4\,\,\,\,\, \Rightarrow \,\,\,\,\,t = 15\left( {L - J} \right) - 2 = 15\left( {L - J - 1} \right) + 15 - 2\,\,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{\\
\,M = L - J - 1 \hfill \cr \\
\,R = 13\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}. \hfill \cr} \right.\)


The correct answer is therefore (D).


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

Regards,
Fabio.