akurathi12 wrote:

What is the remainder when a positive integer ‘P’ is divided by 33?

(1) When P is divided by 11, the remainder is 5.

(2) When P is divided by 99, the remainder is 5.

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

\(P = 33M + R\)

\(M\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,\,,\,\,\,0 \le R\,\,{\mathop{\rm int}} \,\, \le 32\)

\(? = R\)

\(\left( 1 \right)\,\,P = 11K + 5\,\,\,,\,\,\,K\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,\,\,\left\{ \matrix{

\,{\rm{Take}}\,\,K = 0\,\,\,\, \Rightarrow \,\,\,R = 5\,\, \hfill \cr

\,{\rm{Take}}\,\,K = 1\,\,\,\, \Rightarrow \,\,\,R = 16\, \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{INSUFF}}.\)

\(\left( 2 \right)\,\,P = 99L + 5\,\,\,,\,\,\,L\mathop \ge \limits^{\left( * \right)} 0\,\,{\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,\,\,P = 33\left( {3L} \right) + 5\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{

\,M = 3L \hfill \cr

\,? = R = 5\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}{\rm{.}} \hfill \cr} \right.\)

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

Regards,

Fabio.

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