Let S = 1, 2, , 20 . Let R₁ and R₂ be relations on S defined as R₁ = (x, y) : x^2 + y^2 200 and R₂ = (x, y) : x divides y and x y . Then, the number of elements in R₁ R₂ is ____.

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Quantrex Academy · Accepted Answer

Correct answer: 26

The relation R₁ R₂ contains pairs (x, y) from S S such that x^2 + y^2 200 , x divides y , and x y . Since x divides y and x y , we can write y = kx for some integer k 2 . Substituting this into the inequality gives x^2 + (kx)^2 200 , which simplifies to x^2(1 + k^2) 200 . We check the possible values of x S : For x = 1 : 1 + y^2 200 y^2 199 . Since y S , the maximum value of y is 14 . The possible values for y are multiples of 1 starting from 2 , so y 2, 3, , 14 . This gives 13 pairs. For x = 2 : 4 + y^2 200 y^2 19

Let S = 1, 2, , 20 . Let R₁ and R₂ be relations on S defined as R₁ = (x, y) : x^2 + y^2 200 and R₂ = (x, y) : x divides y and x y . Then, the number of elements in R₁ R₂ is ____.

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