Quote:
Two positive integers a and b are divisible by 5, which is their largest common factor. What is the value of a and b?
(1) The lowest POSITIVE number that has both integers a and b as its factors is the product of one of the integers and the greatest common divisor of the two integers.
(2) The smaller integer is divisible by EXACTLY 4 POSITIVE numbers and has the smallest odd prime number as ONE OF its factors.
\(\left. \matrix{\\
a,b\,\, \ge 1\,\,{\rm{ints}} \hfill \cr \\
{\rm{GCD}}\left( {a,b} \right) = 5\,\, \hfill \cr} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{\\
\,a = 5M\,\,,\,\,M \ge 1\,\,{\mathop{\rm int}} \, \hfill \cr \\
\,b = 5N\,\,,\,\,N \ge 1\,\,{\mathop{\rm int}} \, \hfill \cr} \right.\,\,\,\,\,\,M,N\,\,{\rm{relatively}}\,\,{\rm{prime}}\)
\({\rm{?}}\,\,\,{\rm{:}}\,\,\,{\rm{a,}}\,{\rm{b}}\)
Let´s go straight to (1+2), because it is easy to BIFURCATE (1) and (2) together (and this guarantees that each alone is also insufficient):
\(\left( {1 + 2} \right)\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {5 \cdot 3,5 \cdot 5} \right)\,\,\,\,\left( * \right)\, \hfill \cr \\
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {5 \cdot 5,5 \cdot 3} \right)\,\,\,\,\left( {**} \right)\,\, \hfill \cr} \right.\)
\(\left( * \right)\,\,\,\left\{ \matrix{\\
LCM\left( {a,b} \right) = 3 \cdot {5^2} = \left( {5 \cdot 3} \right) \cdot 5 = a \cdot GCD\left( {a,b} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\left( 1 \right)\,\,{\rm{satisfied}} \hfill \cr \\
\,{\rm{5}} \cdot {\rm{3}}\,\,\,{\rm{has}}\,\,{\rm{exactly}}\,\,{\rm{4}}\,\,{\rm{positive}}\,\,{\rm{divisors}}\,\,{\rm{and}}\,\,{\rm{3}}\,\,{\rm{is}}\,\,{\rm{one}}\,\,{\rm{of}}\,\,{\rm{its}}\,\,{\rm{factors}}\,\,\,\,\,\, \Rightarrow \,\,\,\left( 2 \right)\,\,{\rm{satisfied}} \hfill \cr} \right.\)
\(\left( {**} \right)\,\,\,{\rm{analogous}}\,\,\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.