fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 16)
What is the value of the positive two-digit integer N?
(1) N and the product of the digits of N are 12 units apart.
(2) N is greater than the product of the digits of N.
$$? = N = \left\langle {ab} \right\rangle \,\,\,\,\,\,\,\left[ {a \ge 1\,,\,b \ge 0\,\,\,{\rm{digits}}} \right]$$
$$\left( 1 \right)\,\,\,\left| {\left( {10a + b} \right) - ab} \right| = 12\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\left| {\left( {b - 10} \right)\left( {1 - a} \right)} \right| = 2\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\left( {10 - b} \right)\left( {a - 1} \right) = 2$$
$$\,\,\,\,\, \Rightarrow \,\,\,\,\left( {10 - b\,,\,\,a - 1} \right)\,\,{\rm{pair}}\,\,\,{\rm{of}}\,\,{\rm{positive}}\,\,{\rm{divisors}}\,\,{\rm{of}}\,\,2\,\,\,\,\, \Rightarrow \,\,\,\,\left( {10 - b,a - 1} \right) \in \left\{ {\left( {1,2} \right),\left( {2,1} \right)} \right\}$$
$$\left. \matrix{
\left\{ \matrix{
\,10 - b = 1 \hfill \cr
\,a - 1 = 2 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {a,b} \right) = \left( {3,9} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,N = 39\,\,{\rm{viable!}}\,\,\,\, \hfill \cr
\left\{ \matrix{
\,10 - b = 2 \hfill \cr
\,a - 1 = 1 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {a,b} \right) = \left( {2,8} \right)\,\,\,\, \Rightarrow \,\,\,\,\,N = 28\,\,{\rm{viable!}} \hfill \cr} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,N = 28\,\,{\rm{or}}\,\,39$$
$$\left( * \right)\,\,\left( {10a + b} \right) - ab = 12\,\,\,\,\, \Leftrightarrow \,\,\,\,\, - a\left( {b - 10} \right) + b = 12\,\,\,\,\, \Leftrightarrow \,\,\,\,\, - a\left( {b - 10} \right) + b - \underline {10} = 12 - \underline {10} \,\,\,\,\, \Leftrightarrow \,\,\,\,\left( {b - 10} \right)\left( {1 - a} \right) = 2$$
$$\left( {**} \right)\,\,\,\left\{ \matrix{
\,b - 10 < 0\,\,\,\, \Rightarrow \,\,\,\,\left| {b - 10} \right| = 10 - b \hfill \cr
\,1 - a \le 0\,\,\,\, \Rightarrow \,\,\,\,\left| {1 - a} \right| = a - 1 \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\,\left\{ \matrix{
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,N = 28\,\,\,\,\,\left[ {28 > 16} \right]\,\,\,\, \Rightarrow \,\,\,? = 28 \hfill \cr
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,N = 39\,\,\,\,\,\left[ {39 > 27} \right]\,\,\,\, \Rightarrow \,\,\,? = 39 \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left( {\rm{E}} \right)$$
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
P.S.: we know 28 and 39 could be found by some "organized manual work"... but the importance of the solution is contained in the techniques presented, not in the numbers themselves!
