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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