# Tagged Questions

Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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### Does there exist a general technique for solving systems of multivariable linear congruences

I'm aware for coprime moduli we have the CRT for solving the problem $$\begin{matrix} a_0 x \equiv b_0 \mod m_0 \\ a_1 x \equiv b_1 \mod m_1 \\ \vdots \\ a_n x \equiv b_n \mod m_n \end{matrix}$$ ...
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### I need help with the powers of an integer, modulo m

I am currently reading a chapter in math textbook about the powers of an integer, modulo m. I am having troubles with the following claims Suppose that $a^r \equiv a^s(mod\text{ } m)$, where $r>s$...
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### What to keep in mind when attempting proof of basic properties of divisibility/what techniques are useful/what's the intuition for showing them?

So I am currently trying to prove some basic divsiibility relations, as follows. If $a \mid b$ and $a \mid c$, then $a \mid (b + c)$. If $a \mid b$ and $s \in \mathbb{Z}$, then $a \mid sb$. ...
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### ceiling of an expression

If we need to find the ceiling of this expression (A-11)/100 then is it correct to simply write the above expression as ...
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### Product of the first $n$ Fibonacci numbers is a perfect square

Suppose that $F_{n+2}=F_n+F_{n+1}$ and $F_1=F_2=1$. Can the number $P_n=F_1\cdots F_n$ be a perfect square if $n\ge 3$?
### Prove that If $m'$ is a common multiple of $s$ and $t$, then $m | m'$. Here $m$ is the LCM of $s$ and $t$.
Prove that If $m'$ is a common multiple of $s$ and $t$, then $m | m'$. Here $m$ is the LCM of $s$ and $t$. Although the statement is intuitively clear to me I don't know how to prove.
### n-th roots of unity summing to $0$
Let $\zeta = e^{2\pi i/n}$ be an $n$-th root of unity, and let $S = \{\zeta^m|m=0,1,\ldots,n-1\}$ be the corresponding sets of all $n$-th roots of unity. Let $k \leq z$. Let $C \subseteq S$ such ...