query: Let $k, m, n \in\mathbb Z^+$, where $k$ and $m$ are relatively prime. Prove that if $k|mn$ then $k|n$ .
positive_passages.docid: 63183
positive_passages.text: Suppose $a,b,m,n$ are natural numbers, prove that if $a({m \over n})=b$ and $(a,n)=1$, then ${m \over n} \in \mathbb{N}$
negative_passages.docid: 72672
negative_passages.text: Show that if $ar + bs = 1$ for some $r$ and $s$ then $a$ and $b$ are relatively prime
negative_passages.docid: 114867
negative_passages.text: Prove $\forall n\geq 2,n\in\mathbb{Z}$, $(n+1)\mid(n^3+1)$
negative_passages.docid: 34154
negative_passages.text: Prove the statement $\forall x\in\mathbb N (x > 1\to\exists k\in\mathbb N\exists m \in\mathbb N (m \equiv 1 (\text{mod }2) \wedge x = 2^km))$.
negative_passages.docid: 54554
negative_passages.text: If a product of relatively prime integers is an $n$th power, then each is an $n$th power
negative_passages.docid: 46383
negative_passages.text: If $x$ divides $z$, $y$ divides $z$ and $x$ and $y$ are relatively prime, then $xy$ divides $z$.
negative_passages.docid: 27544
negative_passages.text: With $a,\ b,\ m,\ n\in \mathbb{Z}$. Show that if ${\rm gcd}(m,n)=1$, then ${\rm gcd}(ma+nb,mn)={\rm gcd}(a,n){\rm gcd}(b,m)$
negative_passages.docid: 145752
negative_passages.text: Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
negative_passages.docid: 78852
negative_passages.text: If $\gcd(m,n) =1, $ then $m\mathbb{Z}/ mn\mathbb{Z} \cong \mathbb{Z}/(n)$
negative_passages.docid: 19700
negative_passages.text: Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$
negative_passages.docid: 101345
negative_passages.text: Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is...
negative_passages.docid: 148455
negative_passages.text: Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
negative_passages.docid: 8710
negative_passages.text: Bounds for $n$-th prime
negative_passages.docid: 124685
negative_passages.text: How to show $(mn)!$ is divisible by $(m!)^n$?
negative_passages.docid: 51952
negative_passages.text: Show that $x^{n-1}+\cdots +x+1$ is irreducible over $\mathbb Z$ if and only if $n$ is a prime.
negative_passages.docid: 133019
negative_passages.text: Nilpotent elements in $\mathbb Z_n$
negative_passages.docid: 143407
negative_passages.text: Fibonacci Numbers $F_n$ and $F_{n + 1}$ are relatively prime for all $n \geq 0$.