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If \(2^x + 2^y = x^2 + y^2,\) where \(x\) and \(y\) are nonnegative integers, what is the greatest possible value of

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If \(2^x + 2^y = x^2 + y^2,\) where \(x\) and \(y\) are nonnegative integers, what is the greatest possible value of

by Gmat_mission » Fri Aug 20, 2021 11:34 am

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If \(2^x + 2^y = x^2 + y^2,\) where \(x\) and \(y\) are nonnegative integers, what is the greatest possible value of \(|x - y|?\)

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Answer: D

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Re: If \(2^x + 2^y = x^2 + y^2,\) where \(x\) and \(y\) are nonnegative integers, what is the greatest possible value of

by regor60 » Fri Aug 20, 2021 12:39 pm
We know the answer is unlikely to be either 0 or 4 because tests.

But try 4 first.

Either X or Y could be 4 and the other 0, let's test.

2^0 +2^4 =17 but 0^2+4^2=16, so doesn't meet problem statement.

Try 5 and 1.

2^5+2^1=33 but 5^2+1^2= 26, doesn't work either and the difference is increasing.

Increasing X and Y by 1 successively and difference keeps increasing, so it's safe to say 4 doesn't work.

Let's try 3 then.

Set X=0 and Y=3.

2^0+2^3=9. 0^2 + 3^2 also = 9. Problem statement met.

D,3
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