[Math Revolution GMAT math practice question]
A store sold 72 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 store sold 72 watches for $a2,34b, where a2,34b is a 5-dig
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- Max@Math Revolution
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$$? = a + b\,\,\,\,\,\left( {a \ne 0\,\,,\,\,b\,\,\,{\rm{digits}}} \right)\,\,\,\left( * \right)$$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
$${{\left\langle {a234b} \right\rangle } \over {8 \cdot 9}} = {\mathop{\rm int}} \,\,\,\,\,\mathop \Rightarrow \limits^{GCF\left( {8,9} \right)\,\, = \,\,1} \,\,\,\,\,{{\,\left\langle {a234b} \right\rangle } \over 9} = {\mathop{\rm int}} \,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{{a + b + 9} \over 9}\, = {\mathop{\rm int}} \,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{{a + b} \over 9} = {\mathop{\rm int}} \,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,? = a + b = 9$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
POST-MORTEM:
$$\frac{{\left\langle {a234b} \right\rangle }}{{8 \cdot 9}} = \operatorname{int} \,\,\,\,\,\mathop \Rightarrow \limits^{GCF\left( {8,9} \right)\,\, = \,\,1} \,\,\,\,\frac{{\left\langle {a234b} \right\rangle }}{8} = \operatorname{int} \,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\frac{{\left\langle {34b} \right\rangle }}{8}\, = \operatorname{int} \,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\frac{{336 + 4 + b}}{8} = \operatorname{int} \,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\frac{{4 + b}}{8} = \operatorname{int} \,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,b = 4\,\,\,\,\,\,\left( {\therefore \,\,a = 5} \right)\,\,\,$$
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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
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.
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.
- Max@Math Revolution
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=>
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
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
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Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
A store sold 72 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
Assuming the price of each watch is the same, we see that a2,34b must be a multiple of 72. That is, it must be a multiple of 8 and 9 since 72 = 8 x 9. We know that for a number to be a multiple of 8, the last 3 digits must be divisible by 8 and for a number to be a multiple of 9, the sum of the digits must be divisible by 9. So here, we need 34b to be divisible by 8 and a + 2 + 3 + 4 + b = a + b + 9 to be divisible by 9.
If 34b is divisible by 8, then b = 4 only. In that case a must be 5 so that a + b + 9 = 18 will be divisible by 9. So a + b = 5 + 4 = 9.
Answer: E
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