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p(x) is divisible by m - ISl 1968

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September 23, 2010
algebrapolynomialcalculusnumber theoryDivisibilityIMO Shortlist

Problem Statement

A polynomial p(x)=a0xk+a1xk−1+⋯+akp(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k with integer coefficients is said to be divisible by an integer mm if p(x)p(x) is divisible by m for all integers xx. Prove that if p(x)p(x) is divisible by mm, then k!a0k!a_0 is also divisible by mm. Also prove that if a0,k,ma_0, k,m are non-negative integers for which k!a0k!a_0 is divisible by mm, there exists a polynomial p(x)=a0xk+⋯+akp(x) = a_0x^k+\cdots+ a_k divisible by m.m.
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