Prove that $\sqrt{5}$ is irrational. I begin with the identity $(\sqrt{5} + 2 )(\sqrt{5} – 2 ) = 1$. Then I am told to extract $\sqrt{5}$ from the first or second factor and consider it to be $\frac{m}{n}$ so I should replace it in both sides. I have $$\frac{m}{n} = (\frac{1}{\frac{m}{n}} + 2) + 2.$$ […]

This question already has an answer here: $n^5-n$ is divisible by $10$? 7 answers

Suppose we’re given a $k$-long list of rational numbers: $$q : \{0,\ldots,k-1\} \rightarrow \mathbb{Q}.$$ Then there’s a least $n \in \mathbb{Z}_{>0}$ such that $nq_i$ is an integer for all $i$. Question. What is this integer $n$ called, and does it have any generalizations/related concepts in ring theory? For what it’s worth, I think $n$ can […]

I am trying to solve the following problem: find all solutions to the congruence $x^2 \equiv 1$ (mod 91). Already, I have solved the congruence $x^2 \equiv 1$ (mod 7) and (mod 13) and I am trying to use the Chinese Remainder Theorem however I am puzzled by how exactly to use it in this […]

If the sum of the number of divisors:$\ \sum_{n\leq x}d(n)=x\log x+(2\gamma-1)x+O(\sqrt{x}),$ what is $\sum_{n\leq x}d(2n)$ asymptotic to? Is there a generalisation for $\sum_{n\leq x}d(k n),\ k\in \mathbb{N}$?

Find all integer solutions to linear congruences: \begin{align} &(a) &3x &\equiv 24 \pmod{6},\\ &(b) &10x &\equiv 18 \pmod{25},\\ \end{align} What I have so far: $$(a) \gcd(3,6)=3$$ And we know $3|24$ so there are $3$ solutions. By inspection we know that $x=8$ is a solution. One, question I have is, even though $x=8$ is a solution, […]

Does there exist a prime $p \geq 7$ such that the order of $4$ in the multiplicative group of units in $\mathbb{Z}/p^n$ is odd for every positive integer $n$? It would be nice if $7$ was already an example. I computed the order of $4$ modulo $7^n$ for $n=1,2,\ldots,12$ and it came out odd, but […]

$\gcd(n,m)\,{\rm lcm}(n,m) = nm.\,$ Can this theorem work with 3 integers? And how to prove it? I tried doing this with 2 integers n,m , but I can’t figure out how to do it with 3.

Based on the Peano Axioms (wich are a way to correctly absolutely define the set of natural numbers – correct me if i’m wrong) it is possible to construct a set of symbols that doesn’t quite look the way i imagine the natural numbers: If there is a circle of other symbols next to the […]

what will be the remainder when $43$ divides $32002^{4200}$?? what I did is: $32002\equiv10 \pmod{43}$, how to proceed further?

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