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coprime a_{n-1} and a_n - All-Russian MO 2004 Regional (R4) 10.5

Source:

September 27, 2024
algebrapolynomialcoprimeInteger Polynomial

Problem Statement

Equation xn+a1xn−1+a2xn−2+...+an−1x+an=0x^n + a_1x^{n-1} + a_2x^{n-2} +...+ a_{n-1}x + a_n = 0 with integer non-zero coefficients a1a_1, a2a_2, ...... , ana_n has nn different integer roots. Prove that if any two roots are relatively prime, then the numbers an−1a_{n-1} and ana_n are coprime.
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