banksy wrote:

How many prime factors does positive integer n have?

(1) n/5 has only a prime factor.

(2) 3*n^2 has two different prime factors.

VERY nice one!

\(n \ge 1\,\,{\mathop{\rm int}}\)

\(?\,\, = \,\,\# \,\,{\rm{prime}}\,\,{\rm{factors}}\,\,{\rm{of}}\,\,n\)

\(\left( 1 \right)\,\,\,\,\left\{ \matrix{

\,{\rm{Take}}\,\,n = {5^2}\,\,\,\, \Rightarrow \,\,\,\,\,? = 1\,\,\,\,\,\,\,\,\,\,\left( {{\rm{just}}\,\,5} \right)\, \hfill \cr

\,{\rm{Take}}\,\,n = 5 \cdot 2\,\,\,\, \Rightarrow \,\,\,\,\,? = 2\,\,\,\,\,\,\,\left( {2\,\,{\rm{and}}\,\,5} \right) \hfill \cr} \right.\)

\(\left( 2 \right)\,\,\,\,\left\{ \matrix{

\,({\rm{Re}})\,{\rm{Take}}\,\,n = {5^2}\,\,\,\left( {3 \cdot {5^4}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,5} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 1\,\,\,\,\left( {{\rm{just}}\,\,5} \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 = {5^2}\,\,\,\left( {3 \cdot {5^4}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,5} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 1\,\,\,\,\left( {{\rm{just}}\,\,5} \right)\,\,\,\,\,\,\,\,\, \hfill \cr

\,{\rm{Take}}\,\,n = 3 \cdot 5\,\,\,\left( {{3^3} \cdot {5^2}\,\,{\rm{has}}\,\,{\rm{just}}\,\,3\,\,{\rm{and}}\,\,5} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 2\,\,\,\,\,\left( {3\,\,{\rm{and}}\,\,5} \right)\,\,\,\,\,\,\, \hfill \cr} \right.\)

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

Regards,

Fabio.

_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)

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