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Solution to puzzle 95: Integer polynomial

Skip restatement of puzzle.Let P be a polynomial with integer coefficients.  If a, b, c are distinct integers, show that

cannot be satisfied simultaneously.

Suppose, without loss of generality, a < b < c.

By the polynomial remainder theorem, P(c) − P(a) = (c − a)Q(c), for some polynomial Q, with integer coefficients.
Hence c − a divides a − b.
But this is impossible, as |c − a| > |a − b|, and a not equal to b.

Therefore the above system of equations cannot be satisfied simultaneously.

Source: Traditional

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