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k =\frac{x^2 - xy + 2y^2}{x + y} has odd no of ordered integer pairs when 7/k

Source: Czech and Slovak Olympiad 2017, National Round, III A p6

February 2, 2020
number theorydivisible

Problem Statement

Given is a nonzero integer kk. Prove that equation k=x2−xy+2y2x+yk =\frac{x^2 - xy + 2y^2}{x + y} has an odd number of ordered integer pairs (x,y)(x, y) just when kk is divisible by seven.
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