solve-for-x-y-Z-x-y-2020-

Question Number 120185 by mr W last updated on 29/Oct/20

s o l v e f o r x, y \in Z

\sqrt{x} + \sqrt{y} = \sqrt{2020}

Answered by floor(10²Eta[1]) last updated on 29/Oct/20

\sqrt{y} = \sqrt{2020} - \sqrt{x}

(y = 2020 + x - 2 \sqrt{2020 x}) \in Z

\Rightarrow \sqrt{2020 x} \in Z

\sqrt{2020 x} = 2 \sqrt{505 x}

\Rightarrow \sqrt{505 x} \in Z \Rightarrow x = {505 a}^{2}, a \in Z

Similarly : y = {505 b}^{2}, b \in Z

\sqrt{{505 a}^{2}} + \sqrt{{505 b}^{2}} = 2 \sqrt{505}

\Rightarrow a + b = 2

a = 0, b = 2

a = 1, b = 1

a = 2, b = 0

(x, y) \in {(0, 2020), (505, 505), (2020, 0)}

Commented by mr W last updated on 30/Oct/20

t h a n k s!

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