piyush26 wrote:
How many prime factors does the positive integer N have?
(1) The least prime factor of N is greater than √N.
(2) N has exactly 2 positive factors.
Very nice conceptual problem,
piyush26 (kudos) !
\(?\,\, = \,\,\,\,\# \,\,{\text{prime}}\,{\text{factors}}\,\,{\text{of}}\,\,N\,\,\,\,\,\,\left( {N \geqslant 1\,\,\operatorname{int} } \right)\)
\(\left( 1 \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,N\,\,{\text{prime}}\,\,\,\,\, \Rightarrow \,\,\,\,? = 1\,\,\,\left( {N\,\,{\text{itself}}} \right)\)
\(\left( * \right)\,\,\left\{ \begin{gathered}\\
N = 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{no}}\,\,{\text{prime}}\,\,{\text{factors}} \hfill \\\\
N \geqslant 4\,\,{\text{not}}\,\,{\text{prime}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,2 \leqslant p\,\,{\text{prime}}\,\,{\text{factor}}\,\,{\text{of}}\,\,N\,\, \leqslant \sqrt N \,\,{\text{exists}} \hfill \\ \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{contradicts}}\,\,\left( 1 \right)\)
\(\left( 2 \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,N\,\,{\text{prime}}\,\,\,\,\, \Rightarrow \,\,\,\,? = 1\,\,\,\left( {N\,\,{\text{itself}}} \right)\)
\(\left( {**} \right)\,\,\left\{ \begin{gathered}\\
N = 1\,\,\,\,\, \Rightarrow \,\,\,{\text{only}}\,\,1\,\,{\text{positive}}\,\,{\text{factor}}\,\,\left( {N\,\,{\text{itself}} } \right) \hfill \\\\
N \geqslant 4\,\,{\text{not}}\,\,{\text{prime}}\,\,\,\,\, \Rightarrow \,\,\,{\text{more}}\,\,{\text{than}}\,\,{\text{the}}\,\,{\text{trivial}}\,\,{\text{two}}\,\,\left( {1\,\,{\text{and}}\,\,N} \right)\,\,{\text{positive}}\,\,{\text{factors}} \hfill \\ \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{contradicts}}\,\,\left( 2 \right)\,\,\,\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.