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Prove that number of ordered pairs (x,y) is divisible by 3

Source: Turkey TST 2000 P1

March 12, 2011
algebrapolynomialVietanumber theory unsolvednumber theory

Problem Statement

(a)(a) Prove that for every positive integer nn, the number of ordered pairs (x,y)(x, y) of integers satisfying x2−xy+y2=nx^2-xy+y^2 = n is divisible by 3.3. (b)(b) Find all ordered pairs of integers satisfying x2−xy+y2=727.x^2-xy+y^2=727.
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