Find two natural numbers, the sum of whose squares is 25 times their sum and also equal to 50 times their difference.

Let the two natural numbers be x and y.

According to the question

x^{2} + y^{2} = 25(x + y) – – – – – (1)

x^{2} + y^{2} = 50(x – y) – – – – (2)

From (1) and (2) we get

25(x + y) = 50(x – y)

x + y = 2(x – y)

x + y = 2x – 2y

y + 2y = 2x – x

3y = x – – – – – (3)

From (2) and (3) we get

(3y)^{2} + y^{2} = 50(3y – y)

9y^{2} + y^{2} = 50(3y – y)

10 y^{2} = 100y

y = 10

From (3) we have,

x = 3y = 3.10 = 30

Hence the two natural numbers are 30 and 10.

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