BTGmoderatorDC wrote:If a and b are integers such that a > b > 1, which of the following cannot be a multiple of either a or b?

(A) a - 1
(B) b + 1
(C) b - 1
(D) a + b
(E) ab
Source: Manhattan Prep

$$a > b \ge 2\,\,\,{\rm{ints}}$$
$$?\,\,:\,\,\underline {{\rm{not}}} \,\,{\rm{multiple}}\,\,{\rm{of}}\,\,a,b\,$$
$${\rm{Take}}\,\,\left( {a,b} \right) = \left( {3,2} \right)\,\,\,\,\left\{ \matrix{
\,a - 1\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,b\,\,\,\, \Rightarrow \,\,\,\,\left( A \right)\,\,\,{\rm{out}} \hfill \cr
\,b + 1\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,a\,\,\,\, \Rightarrow \,\,\,\,\left( B \right)\,\,\,{\rm{out}} \hfill \cr
\,ab\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,a,b\,\,\,\, \Rightarrow \,\,\,\,\left( E \right)\,\,\,{\rm{out}} \hfill \cr} \right.$$
$${\rm{Take}}\,\,\left( {a,b} \right) = \left( {4,2} \right)\,\,\,\,\left\{ {\,a + b\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,b\,\,\,\, \Rightarrow \,\,\,\,\left( D \right)\,\,\,{\rm{out}}} \right.$$

Conclusion: the correct answer is (C), by exclusion.

Important: from the fact that b-1 is a POSITIVE integer less than both a and b, we are sure b-1 is not a multiple of any one of them!
(-2 is less than both 1 and 2, and -2 is a multiple of both of them. Be careful not to make wrong conclusions!)

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

Regards,
Fabio.