BTGmoderatorDC wrote:If 2 and 17 are factors of positive integer n, then which of the following must also divide into n?
I. 34
II. 68
III. 136
A. I only
B. II only
C. I and II only
D. I and III only
E. None
Source: Manhattan Prep
$$n \ge 1\,\,{\mathop{\rm int}} $$
$$\left\{ \matrix{
{n \over 2} = {\mathop{\rm int}} \hfill \cr
{n \over {17}} = {\mathop{\rm int}} \hfill \cr} \right.\,\,\,\,\,\,\,$$
$$?\,\,\,:\,\,\,{n \over {\,{\rm{I}}\,{\rm{,}}\,{\rm{II}}\,{\rm{,}}\,{\rm{III}}\,}}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}} $$
$${\rm{I}}.\,\,\,\,{n \over {34}}\,\, = \,\,{n \over {2 \cdot 17}}\,\,\mathop = \limits^{GCF\,\left( {2,17} \right)\,\, = \,\,1} \,\,\,\,{\mathop{\rm int}} \,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,$$
$${\rm{II}}.\,\,\,\,{n \over {68}}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}} \,\,\,\,,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\left[ {\,{\rm{Take}}\,\,n = 34\,} \right]$$
$${\rm{III}}.\,\,\,\,{n \over {136}}\,\,\,\mathop = \limits^? \,\,\,{\mathop{\rm int}} \,\,\,\,,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\left[ {\,{\rm{Take}}\,\,n = 34\,} \right]$$
$$\left( {\,\,\left\langle {{\rm{NO}}} \right\rangle = \,\,{\rm{not}}\,\,{\rm{necessarily}}\,} \right)$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
