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Part of 1969 IMO Shortlist

Problems(1)

Polynomial and divisibility

Source:

9/29/2010
(GBR1)(GBR 1) The polynomial P(x)=a0xk+a1xk−1+⋯+akP(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k, where a0,⋯ ,aka_0,\cdots, a_k are integers, is said to be divisible by an integer mm if P(x)P(x) is a multiple of mm for every integral value of xx. Show that if P(x)P(x) is divisible by mm, then a0⋅k!a_0 \cdot k! is a multiple of mm. Also prove that if a,k,ma, k,m are positive integers such that ak!ak! is a multiple of mm, then a polynomial P(x)P(x) with leading term axkax^kcan be found that is divisible by m.m.
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