When A and B are positive integers, is AB a multiple of 4?

Another beautiful problem, Max. Congrats! (and kudos!)

\(A,B\,\,\, \ge 1\,\,\,{\rm{ints}}\)

\({{A \cdot B} \over 4}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}}\)

\(\left( 1 \right)\,\,\,GCD\left( {A,B} \right) = 6\,\,\,\, \Rightarrow \,\,\,\left\{ \matrix{\\
\,A = 6M\,,\,\,\,M \ge 1\,\,{\mathop{\rm int}} \hfill \cr \\
\,B = 6N\,,\,\,\,N \ge 1\,\,{\mathop{\rm int}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,{\rm{with}}\,\,\,\,M,N\,\,\,{\rm{relatively}}\,\,\,{\rm{prime}}\)

\(?\,\,\,\,:\,\,\,\,{{A \cdot B} \over 4}\,\,\, = \,\,\,{{6M \cdot 6N} \over 4}\,\,\, = \,\,\,9MN\,\,\, = \,\,\,{\mathop{\rm int}} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle\)


\(\left( 2 \right)\,\,\,LCM\left( {A,B} \right) = 30\,\,\,\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {A,B} \right) = \left( {1,30} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr \\
\,{\rm{Take}}\,\,\left( {A,B} \right) = \left( {2,30} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.\)


The correct answer is (A), indeed.


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

Regards,
Fabio.