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.
Answer: [spoiler]____(E)__[/spoiler]
What is the value of the positive two-digit integer N?
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- fskilnik@GMATH
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$$? = N = \left\langle {ab} \right\rangle \,\,\,\,\,\,\,\left[ {a \ge 1\,,\,b \ge 0\,\,\,{\rm{digits}}} \right]$$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.
$$\left( 1 \right)\,\,\,\left| {\left( {10a + b} \right) - ab} \right| = 12\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\left| {\left( {b - 10} \right)\left( {1 - a} \right)} \right| = 2\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\left( {10 - b} \right)\left( {a - 1} \right) = 2$$
$$\,\,\,\,\, \Rightarrow \,\,\,\,\left( {10 - b\,,\,\,a - 1} \right)\,\,{\rm{pair}}\,\,\,{\rm{of}}\,\,{\rm{positive}}\,\,{\rm{divisors}}\,\,{\rm{of}}\,\,2\,\,\,\,\, \Rightarrow \,\,\,\,\left( {10 - b,a - 1} \right) \in \left\{ {\left( {1,2} \right),\left( {2,1} \right)} \right\}$$
$$\left. \matrix{
\left\{ \matrix{
\,10 - b = 1 \hfill \cr
\,a - 1 = 2 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {a,b} \right) = \left( {3,9} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,N = 39\,\,{\rm{viable!}}\,\,\,\, \hfill \cr
\left\{ \matrix{
\,10 - b = 2 \hfill \cr
\,a - 1 = 1 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {a,b} \right) = \left( {2,8} \right)\,\,\,\, \Rightarrow \,\,\,\,\,N = 28\,\,{\rm{viable!}} \hfill \cr} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,N = 28\,\,{\rm{or}}\,\,39$$
$$\left( * \right)\,\,\left( {10a + b} \right) - ab = 12\,\,\,\,\, \Leftrightarrow \,\,\,\,\, - a\left( {b - 10} \right) + b = 12\,\,\,\,\, \Leftrightarrow \,\,\,\,\, - a\left( {b - 10} \right) + b - \underline {10} = 12 - \underline {10} \,\,\,\,\, \Leftrightarrow \,\,\,\,\left( {b - 10} \right)\left( {1 - a} \right) = 2$$
$$\left( {**} \right)\,\,\,\left\{ \matrix{
\,b - 10 < 0\,\,\,\, \Rightarrow \,\,\,\,\left| {b - 10} \right| = 10 - b \hfill \cr
\,1 - a \le 0\,\,\,\, \Rightarrow \,\,\,\,\left| {1 - a} \right| = a - 1 \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\,\left\{ \matrix{
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,N = 28\,\,\,\,\,\left[ {28 > 16} \right]\,\,\,\, \Rightarrow \,\,\,? = 28 \hfill \cr
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,N = 39\,\,\,\,\,\left[ {39 > 27} \right]\,\,\,\, \Rightarrow \,\,\,? = 39 \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left( {\rm{E}} \right)$$
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!
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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