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Congruence equation

Source: Chinese TST 2009 4th P2

April 5, 2009
algebrapolynomialmodular arithmeticnumber theoryprime numbersnumber theory proposed

Problem Statement

Find all integers n≥2 n\ge 2 having the following property: for any k k integers a1,a2,⋯ ,ak a_{1},a_{2},\cdots,a_{k} which aren't congruent to each other (modulo n n), there exists an integer polynomial f(x) f(x) such that congruence equation f(x)≡0(modn) f(x)\equiv 0 (mod n) exactly has k k roots x≡a1,a2,⋯ ,ak(modn). x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).
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