swerve wrote:If a is a positive integer and 81 divided by a results in a remainder of 1, what is the value of a?

1) The remainder when a is divided by 40 is 0.
2) The remainder when 40 is divided by a is 40.
Source: Manhattan Prep

Very nice conceptual problem!

$$a \ge 1\,\,{\mathop{\rm int}} \,\,\,\,\,\left( * \right)$$
$$81 = M \cdot a + 1\,\,,\,\,M\mathop \ge \limits^{\left( * \right)} 1\,\,{\mathop{\rm int}} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,M \cdot a = 80\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,a \le 80\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{positive}}\,\,{\rm{divisor}}\,\,{\rm{of}}\,\,80\,\,\,\,\left( {**} \right)$$
$$? = a$$

$$\left( 1 \right)\,\,a = 40J\,\,,\,\,\,J\mathop \ge \limits^{\left( * \right)} 1\,\,\,{\mathop{\rm int}} \left\{ \matrix{
\,{\rm{Take}}\,\,J = 1\,\,\,\,\left( {M = 2} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 40\,\,\,\,{\rm{viable}} \hfill \cr
\,{\rm{Take}}\,\,J = 2\,\,\,\,\left( {M = 1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 80\,\,\,\,{\rm{viable}} \hfill \cr} \right.$$

$$\left( 2 \right)\,\,40 = N \cdot a + \underline {40} \,\,,\,\,N\,\,{\mathop{\rm int}} \,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\left\{ \matrix{
\,a\,\, > \,\,\underline {40} \hfill \cr
\,N = 0 \hfill \cr} \right.\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,a = 80\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{\rm{SUFF}}.$$


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.