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If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1)

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marcusaurelius
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If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink] New post   Updated on: 18 Jun 2018, 04:31
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If \((2^x)(3^y) = 288\), where x and y are positive integers, then \((2^{x-1})(3^{y-2})\) equals:

A. 16
B. 24
C. 48
D. 96
E. 144
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A

Originally posted by marcusaurelius on 21 May 2010, 11:33.
Last edited by Bunuel on 18 Jun 2018, 04:31, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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skipjames
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Re: If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink] New post   21 May 2010, 11:51
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If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1)(3^y-2) equals:

16
24
48
96
144


So I would start attacking this problem by quickly performing the prime factorization of 288. With that it is easy to count the 5 twos and the 2 threes that are the prime factors. So x=5, y=2. now quickly 2^4(3^0)=16. Than answer should be number 1.

If my comments were helpful to you, please give kudos.
Thanks,
Skip
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Re: If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink] New post   15 Jun 2010, 04:37
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Algebraically we can approach the problem in another way as well.

If (2^x)(3^y) = 288
then (2^x-1)(3^y-2) = 288/(2[*]3[*]3)
The reason being the exponent of 2 is reduced by 1 and exponent of 3 is reduced by 2.
Hence, the answer would be 288/18 = 16.

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marcusaurelius
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Re: If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink] New post   21 May 2010, 11:54
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Thanks, I didn't even think to use the prime factorization.
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Re: If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink] New post   12 Nov 2011, 18:45
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skipjames wrote:
If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1)(3^y-2) equals:

16
24
48
96
144


So I would start attacking this problem by quickly performing the prime factorization of 288. With that it is easy to count the 5 twos and the 2 threes that are the prime factors. So x=5, y=2. now quickly 2^4(3^0)=16. Than answer should be number 1.

If my comments were helpful to you, please give kudos.
Thanks,
Skip


Hi Skip, could you please explain prime factorization and how you count the three's. I think I see how you count the two's by taking 2, 8, and 8 seperately and see how many two's making up each number but I don't quite understand how you get the three's.

Thanks, Steven
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Re: If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink] New post   14 Nov 2011, 14:06
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Here's how I do prime factorization.

In one row you start with your number to be factored and in the other column you write the smallest possible prime that number is divisible by. Then you do the same for the (number/prime) and so on until you get to 1.

For example:

number:______288______144______72______36______18______9______3______1 (done)
prime factor:_______2_________2______ 2_______2_______2______3______3

In this case you it's easy to see that 288 = 2*2*2*2*2*3*3 = 2^5 * 3^2.
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Re: If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink] New post   15 Nov 2011, 09:27
I approached it by using prime factorization. you need to know how many two's and three's are there in 288. using prime factorization, i got 4 two's and 2 three's.
you just need to plug in the values for x and y and solve the given equation.
(2^5-1)(3^2-2)= (2^4)(3^0)= (16)(1)= 16

hope that helps
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Re: If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink] New post   29 Apr 2017, 04:25
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Prime factorize 288 then do your magic:

\(\left( { 2 }^{ x } \right) \left( 3^{ y } \right) =288\\ \left( { 2 }^{ x } \right) \left( 3^{ y } \right) ={ 2 }^{ 5 }{ 3 }^{ 2 }\\ x=5\\ y=2\\ \\ \left( { 2 }^{ x-1 } \right) \left( { 3 }^{ y-2 } \right) =\left( { 2 }^{ 5-1 } \right) \left( { 3 }^{ 2-2 } \right) ={ 2 }^{ 4 }{ 3 }^{ 0 }=16\)
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Re: If 2^x*3^y = 288, where x and y are positive integers then 2^(x-1)*3 [#permalink] New post   18 Dec 2018, 03:02
Bunuel wrote:
If \(2^x*3^y = 288\), where x and y are positive integers then \(2^{x-1}*3^{y-2}\) is equal to?

A. 2
B. 4
C. 6
D. 8
E. 16


\(288 = 3 *96 = 3 * 3*32 = 3*3* 4*8 = 3*3 * 2*2 * 2*2*2 = 3^22^5\)

\(2^x*3^y = 288\)

\(2^x3^y = 2^53^2\)

x = 5 and y = 2.

\(2^{x-1}*3^{y-2}\)

=\(2^{5-1} *3^{2-2}\)

=\(2^4 * 3^0\)

=16

E is our answer.
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Re: If 2^x*3^y = 288, where x and y are positive integers then 2^(x-1)*3 [#permalink] New post   21 Dec 2018, 07:52
Break down 288 into its prime factors. You find that X = 5 and Y = 2. The rest is...
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Re: If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink] New post   22 Dec 2018, 00:19
288= 2*2*2*2*2*3*3 = 2^5 * 3^2

(2^x-1) * (3^y-2) = 2^4 * 3^2-2
= 16 *1= 16
Answer A.
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Re: If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink] New post   05 Oct 2020, 03:22
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If we break up the term as the following then \(2^{x-1} * 3^{y-2} = \frac{2^x}{2} * \frac{3^y}{3^2}\)

Since \(2^{x} * 3^{y} = 288\), substituting, we get \(\frac{288}{2 * 9} = \frac{32}{2} = 16\)


Option A

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Re: If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink] New post   05 May 2022, 06:28
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marcusaurelius wrote:
If \((2^x)(3^y) = 288\), where x and y are positive integers, then \((2^{x-1})(3^{y-2})\) equals:

A. 16
B. 24
C. 48
D. 96
E. 144


APPROACH #1: Use prime factorization
\(288 = (2)(2)(2)(2)(2)(3)(3) = (2^5)(3^2)\)
So, the given equation becomes: \((2^x)(3^y) = (2^5)(3^2)\), which means \(x=5\) and \(y=2\)
Thus \((2^{x-1})(3^{y-2}) = (2^{5-1})(3^{2-2}) = (2^{4})(3^{0}) = (16)(1) = 16\)
Answer: A

APPROACH #2: Use algebraic manipulation
Given: \((2^x)(3^y) = 288\)
Divide both sides of the equation by \(2^1\) (aka \(2\)) to get: \(\frac{(2^x)(3^y)}{2^1} = \frac{288}{2^1}\)
Simplify to get: \((2^{x-1})(3^y) = 144\)
Divide both sides of the equation by \(3^2\) to get: \(\frac{(2^{x-1})(3^y)}{3^2} = \frac{144}{3^2}\)
Simplify: \((2^{x-1})(3^{y-2}) = 16\)
Answer: A
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Re: If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink] New post   20 Apr 2024, 03:13
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Re: If (2^x)(3^y) = 288, where x and y are positive integers, then (2^x-1) [#permalink]
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