BTGmoderatorDC wrote:How many prime factors does positive integer n have?
(1) n/7 has only one prime factor.
(2) 3*n^2 has two different prime factors.
Source: Veritas Prep
Nice one!
$$n \ge 1\,\,{\mathop{\rm int}} $$
$$?\,\, = \,\,\# \,\,{\rm{prime}}\,\,{\rm{factors}}\,\,{\rm{of}}\,\,n$$
$$\left( 1 \right)\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,n = {7^2}\,\,\,\, \Rightarrow \,\,\,\,\,? = 1\,\,\,\,\,\,\,\,\,\,\left( {{\rm{just}}\,\,7} \right)\, \hfill \cr
\,{\rm{Take}}\,\,n = 7 \cdot 2\,\,\,\, \Rightarrow \,\,\,\,\,? = 2\,\,\,\,\,\,\,\left( {2\,\,{\rm{and}}\,\,7} \right) \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\,\,\left\{ \matrix{
\,({\rm{Re}})\,{\rm{Take}}\,\,n = {7^2}\,\,\,\left( {3 \cdot {7^4}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,7} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 1\,\,\,\,\left( {{\rm{just}}\,\,7} \right)\,\,\,\,\,\,\,\,\, \hfill \cr
\,{\rm{Take}}\,\,n = 2 \cdot 3\,\,\,\left( {{3^3} \cdot {2^2}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,2} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2\,\,\,\,\,\left( {2\,\,{\rm{and}}\,\,3} \right)\,\,\,\,\,\,\, \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,\left\{ \matrix{
\,({\rm{Re}})\,{\rm{Take}}\,\,n = {7^2}\,\,\,\left( {3 \cdot {7^4}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,7} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 1\,\,\,\,\left( {{\rm{just}}\,\,7} \right)\,\,\,\,\,\,\,\,\, \hfill \cr
\,{\rm{Take}}\,\,n = 3 \cdot 7\,\,\,\left( {{3^3} \cdot {7^2}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,7} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2\,\,\,\,\,\left( {3\,\,{\rm{and}}\,\,7} \right)\,\,\,\,\,\,\, \hfill \cr} \right.$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
