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Polynomials, Integers and Divisibility

Problem 6. Suppose that f(s) = adxd+ad-1xd... is a polynomial with integral coefficients (so a0, a1 . . ,ad E Z). Show that
f(n)?f(m) is divisible by n ? m for all distinct integers n and m.


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Ok, the main thing to observe here is that fact that: for any integer like p and distinct integers like m and n, (n^p-m^p) is divisible by (n-m). In fact we always have that:


That can be verified by direct division i.e. ...

Solution Summary

Polynomials, Integers and Divisibility are investigated.