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Let $f(x)=(x+b)^2-c$, where $b$ and $c$ are integers.

(a) If $p$ is a prime number such that $c$ is divisible by $p$, but not by $p^2$, show that $p^2$ does not divide $f(n)$ for any integer $n$.

(b) Let $q \neq 2$ be a prime divisor of $c$. If $q$ divides $f(n)$ for some integer $n$, prove that for every positive integer $r$ there exists an integer $n^{\prime}$ for which $q^r$ divides $f\left(n^{\prime}\right)$
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