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If (2^a)*3^(b-1) = (18^b)/2 where a and b are integers, what is the v

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If (2^a)*3^(b-1) = (18^b)/2 where a and b are integers, what is the v  [#permalink]

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New post Updated on: 28 Feb 2019, 13:25
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GMATH practice exercise (Quant Class 3)

If \({2^a} \cdot {3^{b - 1}} = {{{{18}^b}} \over 2}\) , where \(a\) and \(b\) are integers, what is the value of \(ab\) ?

(A) -3
(B) -1
(C) 0
(D) 1
(E) 2

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net

Originally posted by fskilnik on 28 Feb 2019, 08:36.
Last edited by fskilnik on 28 Feb 2019, 13:25, edited 1 time in total.
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Re: If (2^a)*3^(b-1) = (18^b)/2 where a and b are integers, what is the v  [#permalink]

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New post 28 Feb 2019, 09:09
IMO E .

2^a)*3^(b-1) = (18^b)/2,
= 2^b-1.3^2b

Cmpare the powers of 2 and 3 , we get

a=b-1 and b-1=2b
=> b=-1 and a = -2

So ab = 2

E.
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Re: If (2^a)*3^(b-1) = (18^b)/2 where a and b are integers, what is the v  [#permalink]

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New post 28 Feb 2019, 11:15
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fskilnik wrote:
GMATH practice exercise (Quant Class 3)

If \({2^a} \cdot {3^{b - 1}} = {{{{18}^b}} \over 2}\) , what is the value of \(ab\) ?

(A) -3
(B) -1
(C) 0
(D) 1
(E) 2


\({2^a} \cdot {3^{b - 1}} = {{{{18}^b}} \over 2}\)

2^a*3^b-1= 3^2b*2^b-1

a=b-1
b-1=2b
b=-1
so
a=-2
ab= 2
IMO E
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Re: If (2^a)*3^(b-1) = (18^b)/2 where a and b are integers, what is the v  [#permalink]

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New post 28 Feb 2019, 13:24
fskilnik wrote:
GMATH practice exercise (Quant Class 3)

If \({2^a} \cdot {3^{b - 1}} = {{{{18}^b}} \over 2}\) , where \(a\) and \(b\) are integers, what is the value of \(ab\) ?

(A) -3
(B) -1
(C) 0
(D) 1
(E) 2

\(? = \,\,ab\)

\({2^a} \cdot {3^{b - 1}} = {{{{18}^b}} \over 2}\,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,2} \,\,\,\,\,{2^{a + 1}} \cdot {3^{b - 1}}\, = {\left( {2 \cdot {3^2}} \right)^b} = {2^b} \cdot {3^{2b}}\)

\({2^{a + 1 - b}} = {3^{2b - \left( {b - 1} \right)}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\left\{ \matrix{
\,a + 1 - b = 0 \hfill \cr
\,2b - \left( {b - 1} \right) = 0\, \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {a,b} \right) = \left( { - 2, - 1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 2\)

\(\left( * \right)\,\,\,{\rm{integer}}\,\,{\rm{exponents}}\)


The correct answer is (E).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: If (2^a)*3^(b-1) = (18^b)/2 where a and b are integers, what is the v   [#permalink] 28 Feb 2019, 13:24
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