Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
A store sold 72 identical watches for $a2,34b, where a2,34b is a 5-digit integer. What is the value of a + b?
A. 5
B. 6
C. 7
D. 8
E. 9
$$? = a + b\,\,\,\,\,\left( {a \ne 0\,\,,\,\,b\,\,\,{\rm{digits}}} \right)\,\,\,\left( * \right)$$
$${{\left\langle {a234b} \right\rangle } \over {8 \cdot 9}} = {\mathop{\rm int}} \,\,\,\,\,\mathop \Rightarrow \limits^{GCF\left( {8,9} \right)\,\, = \,\,1} \,\,\,\,\,{{\,\left\langle {a234b} \right\rangle } \over 9} = {\mathop{\rm int}} \,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{{a + b + 9} \over 9}\, = {\mathop{\rm int}} \,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{{a + b} \over 9} = {\mathop{\rm int}} \,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,? = a + b = 9$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
POST-MORTEM:
$$\frac{{\left\langle {a234b} \right\rangle }}{{8 \cdot 9}} = \operatorname{int} \,\,\,\,\,\mathop \Rightarrow \limits^{GCF\left( {8,9} \right)\,\, = \,\,1} \,\,\,\,\frac{{\left\langle {a234b} \right\rangle }}{8} = \operatorname{int} \,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\frac{{\left\langle {34b} \right\rangle }}{8}\, = \operatorname{int} \,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\frac{{336 + 4 + b}}{8} = \operatorname{int} \,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\frac{{4 + b}}{8} = \operatorname{int} \,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,b = 4\,\,\,\,\,\,\left( {\therefore \,\,a = 5} \right)\,\,\,$$
