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If a, b, x, y are positive integers and a^xb^y= 144, then a + b = ?

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If a, b, x, y are positive integers and a^xb^y= 144, then a + b = ? [#permalink]

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If a, b, x, y are positive integers and \(a^{x}b^y= 144\), then a + b = ?

1. x > y

2. xy = 8
[Reveal] Spoiler: OA

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If a, b, x, y are positive integers and a^xb^y= 144, then a + b = ? [#permalink]

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idontknowwhy94 wrote:
If a, b, x, y are positive integers and \(a^{x}b^y= 144\), then a + b = ?

1. x > y

2. xy = 8

Dear idontknowwhy94,

I'm happy to respond. :-)

Here, it's helpful to take the prime factorization of 144:
144 = 12*12 = 3*4*3*4 = (3^2)(4^2) = (3^2)(2^4)

Statement #1: x > y
The exponent of a must be larger. Certainly this could be (2^4)(3^2), so that x = 4, y = 2, and x > y. Thus, x + y = 6
Remember also, that if b = 12, and y = 2, then a would have to equal 1, and x could be any power on earth.
We could have (1^137)*(12^2), so that x = 137, y = 2, and x > y. This also works, but x + y = 139, a different answer.
Two different choices, two answer to the prompt question. This statement, alone and by itself, is insufficient.

Statement #2: xy = 8
This is an interesting restriction.
Case i: we still can use (2^4)(3^2), so that xy = 8. Again, this gives x + y = 6.
Case ii: suppose b = 144. Then y = 1, a = 1, and we could pick x = 8.
144 = (1^8)(144^1), and this also give xy = 8, so this is also valid. This gives x + y = 9.
Once again, two different choices give two answer to the prompt question. This statement, alone and by itself, is also insufficient.

Combined:
The problem is, the two cases we used for statement #2 also worked for statement #1.
Case i: 144 = (2^4)(3^2)
Case ii: 144 = (1^8)(144^1)
Both of these scenarios satisfy both statements, but they give different sums for x + y. Even with all the information, we cannot determine a unique answer to the prompt question. Even with both statements, everything is insufficient.

Answer = (E)

Does all this make sense?
Mike :-)
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Magoosh Test Prep

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Re: If a, b, x, y are positive integers and a^xb^y= 144, then a + b = ? [#permalink]

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Re: If a, b, x, y are positive integers and a^xb^y= 144, then a + b = ?   [#permalink] 19 Oct 2017, 19:16
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