Bunuel 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

\(a > b \ge 2\,\,\,{\rm{ints}}\)

\(?\,\,:\,\,\underline {{\rm{not}}} \,\,{\rm{multiple}}\,\,{\rm{of}}\,\,a,b\,\)

\({\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.

_________________

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

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