2009/10 AHSMC Part 1 Problem 11

Problem 11 of 16

$Q(x)$ is a polynomial with integer coefficients such that $Q(9) = 2009$ . If $p$ is a prime number such that $Q(p) = 392$ , then $p$ can

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Since $Q(9) = 2009$ , $x - 9$ is a factor of $Q(x) - 2009$ . Since $Q(x)$ has integral coefficients, $p - 9$ divides $Q(p) - 2009 = 392 - 2009$ . Since this number is odd, $p - 9$ must be odd. Since $p$ is prime, we must have $p = 2$ . Now $392 - 2009 = -1617 = -231 \times 7$ , so that $Q(x)$ does exist. For instance, we may have $Q(x) = 231x - 70$ . The answer is (A).

University of Alberta

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