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:
$$\frac{{\left\langle {a234b} \right\rangle }}{{8 \cdot 9}} = \operatorname{int} \,\,\,\,\,\mathop \Rightarrow \limits^{GCF\left( {8,9} \right)\,\, = \,\,1} \,\,\,\,\frac{{\left\langle {a234b} \right\rangle }}{8} = \operatorname{int} \,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\frac{{\left\langle {34b} \right\rangle }}{8}\, = \operatorname{int} \,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\frac{{336 + 4 + b}}{8} = \operatorname{int} \,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\frac{{4 + b}}{8} = \operatorname{int} \,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,b = 4\,\,\,\,\,\,\left( {\therefore \,\,a = 5} \right)\,\,\,$$

Answer: E
Since 72 identical watches were sold, the no. a234b must be divisible by 72.
Since 72=8*9,For any no. to be divisible by 72, it must be divisible by 8 and 9.
This means sum of all the digits of the no., a+2+3+4+b should be divisible by 9, which implies a+b+9 should be divisible by 9.
Out of the given options, only when a+b=9, will the sum of digits of the no. will be 18,which is divisible by 9.Hence the answer

so 72 * price of each watch =a234b
so price of each=a234b/72


for numerator to div by 72, it must be divisible by all factors of 72.

now 72=2^3*3^2=6*9
so in the denominator, we have 9

now for numerator to be divisible by 9, the sum of digits must be divisible by 9.(divisibility rule of 9)
so a+2+3+4+b=9+a+b
so if a+b is 9
it becomes 18 which is div by 9

so, E is the correct answer.

=>

Since a234b is a multiple of 72, a234b is a multiple of both 8 and 9.
The last three digits 34b of a234b form a multiple of 8. So, we must have b = 4 since 344 is the only 3-digit multiple of 8 beginning with the digits, 34.
Since a234b is a multiple of 9, a + 2 + 3 + 4 + b = a + 2 + 3 + 4 + 4 = a + 13 is a multiple of 9. This implies that a = 5, and a + b = 9.

Therefore, E is the answer.
Answer: E