The number of positive integers not greater than 100, which are not di

Hi, Bunuel. Beautiful problem!




\(? = {\rm{Remainder}}\)


\(\left. \matrix{\\
\# \,\,{\rm{div}}\,\,{\rm{by}}\,\,2\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over 2}} \right\rfloor = 50 \hfill \cr \\
\# \,\,{\rm{div}}\,\,{\rm{by}}\,\,3\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over 3}} \right\rfloor = {{99} \over 3} = 33 \hfill \cr \\
\# \,\,{\rm{div}}\,\,{\rm{by}}\,\,5\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over 5}} \right\rfloor = 20 \hfill \cr \\
\# \,\,{\rm{div}}\,\,{\rm{by}}\,\,2\,\,{\rm{and}}\,\,3\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over 6}} \right\rfloor = {{96} \over 6} = 16 \hfill \cr \\
\# \,\,{\rm{div}}\,\,{\rm{by}}\,\,2\,\,{\rm{and}}\,\,5\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over {10}}} \right\rfloor = 10 \hfill \cr \\
\# \,\,{\rm{div}}\,\,{\rm{by}}\,\,3\,\,{\rm{and}}\,\,5\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over {15}}} \right\rfloor = {{90} \over {15}} = 6 \hfill \cr \\
\# \,\,{\rm{div}}\,\,{\rm{by}}\,\,2\,\,{\rm{and}}\,\,3\,\,{\rm{and}}\,\,5\,\,\,:\,\,\,\,\,\left\lfloor {{{100} \over {30}}} \right\rfloor = {{90} \over {30}} = 3\,\,\, \hfill \cr} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,A \cup B \cup C\,\,\mathop = \limits^{{\rm{simplifier}}} \,\,50 + 33 + 20 - \left( {7 + 13 + 3} \right) - 2 \cdot 3 = 74\)


\(? = 100 - 74 = 26\)


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.