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How do I show that an element is in $\mathbb{Z}/p\mathbb{Z}$ given two of its powers?

I have that an element $x$ is in the algebraic closure of $\mathbb{Z}/p\mathbb{Z}$ with $x^m$ and $x^n$ given, and we know $(m, n)=1$. I suppose to show that $x$ is in $\mathbb{Z}/p\mathbb{Z}$ I ...
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expressing contour integral in different form

Hi I have a short question regarding contour integration: Given that $f(z)$ is a continuous function over a rectifiable contour $z = x + iy$. If $f(z) = u(x,y) + iv(x,y)$, why does it follow that the ...
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Spencer and Shelah zero-one law for Erdos-Renyi random graph $G(n,p)$

In Erdos-Renyi random graph $G(n,p(n))$; set $p(n)= (\frac{ln n}{n})^2$. We know that already Spencer and Shelah have proved that zero-one law doesn't hold for $p(n)= \frac{ln n}{n}$. Now the question ...
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Find k if given the constant term of a binomial expression?

Consider the expansion of $x^2(3x^2+\frac{k}{x})^8$. The constant term is 16,128. Find k. This is simply an example of a type of question I cannot understand how to do. I have many questions: 1) ...