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For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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New post 12 Jun 2018, 11:09
Bunuel wrote:
For integers x and y, \(2^x + 2^y=2^{30}\). What is the value of \(x + y\)?

A. 30
B. 32
C. 46
D. 58
E. 64


Dividing both sides by 2^x, we have:

1 + 2^(y - x) = 2^(30 - x)

We see that the right hand side is an integer power of 2. But the left hand side is 1 more than an integer power of 2, which can happen only if 2^(y - x) is also equal to 1. (When that happens, then the left side becomes 1 + 1 = 2, which is an integer power of 2.) (Also note that the only way in which 2^(y - x) can equal 1 is if (x - y) = 0.) Thus, the two sides of the equation can’t be equal unless y - x = 0, so that the left hand side is also an integer power of 2. So we have

1 + 2^0 = 2^(30 - x)

1 + 1 = 2^(30 - x)

2^1 = 2^(30 - x)

1 = 30 - x

x = 29

Since y - x = 0, so y = 29. Thus x + y = 29 + 29 = 58.

Answer: D
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

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New post 06 Feb 2019, 23:06
Bunuel wrote:
For integers x and y, \(2^x + 2^y=2^{30}\). What is the value of \(x + y\)?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.


keyword: test small numbers

\(2^4 = 2^x + 2^y\)

How is that possible, only when x = y = 3

\(2^5 = 2^x + 2^y\), 32 = \(2^4 + 2^4\), here as well x =y = 4

So to get \(2^30\), x and y have to one less than 30 and x=y = 29

58

D
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Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?   [#permalink] 06 Feb 2019, 23:06

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