Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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Increasing sequence of divisors of a quadratic trinomial

This question is from a Russian contest, the 2011 Tuymaada Olympiad. It's the fourth question on day two for the problems at grade level 2. Let $P(n)$ be a quadratic trinomial with integer ...
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Show that the number of reduced residues $a \mod m$ such that $a^{m-1} \equiv 1 \mod m$ is exactly $\displaystyle \prod_{p \mid m} \gcd(p-1,m-1).$

Show that the number of reduced residues $a \mod m$ such that $a^{m-1} \equiv 1 \mod m$ is exactly $$\displaystyle \prod_{p \mid m} \gcd(p-1,m-1).$$ Suppose $f(x) = x^{m−1}−1$ and let $m = ...
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Without using Heegner-Stark-Baker, $\mathbb{Q}(\sqrt{-11})$ has class number $1$.

Prove that $\mathbb{Q}(\sqrt{-11})$ is of class number $1$. I have found that the ideal $(2)$ of the integer ring $\mathbb{Z}[(1 + \sqrt{-11})/2]$ of $\mathbb{Q}(\sqrt{-11})$ is a prime ideal. ...