query: Let $a$ and $b$ commute. If $m$ and $n$ are relatively prime, then ord($ab$) = $mn$.
positive_passages.docid: 74187
positive_passages.text: $\operatorname{ord}(u)=r,\,\operatorname{ord}(v)=s$ $\,\Rightarrow\,\operatorname{ord}(uv)=rs\,$ if $\,r,s\,$ (co)primes
negative_passages.docid: 100426
negative_passages.text: Let $a$ and $b$ be integers. Show that if $a \equiv b \pmod{p}$ for every prime $p$, then $a = b$.
negative_passages.docid: 422
negative_passages.text: If $\gcd(a,b)=1$, $\gcd(a,y)=1$ and $\gcd(b,x)=1$ then prove that $ax+by$ is prime to $ab$
negative_passages.docid: 119051
negative_passages.text: Show that only finitely many positive integers $n$ have the property that for all $m$ for which $1<m<n$ and $\text{gcd}(m,n) = 1$, $m$ is prime.
negative_passages.docid: 156443
negative_passages.text: if $x$ and $y$ commute with $[x y]$, then $[x y]$ is nilpotent
negative_passages.docid: 37061
negative_passages.text: The order of $ab$ when $a,b$ commute
negative_passages.docid: 66748
negative_passages.text: Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
negative_passages.docid: 32344
negative_passages.text: Let $a,k,m$ be integers. Prove that $\gcd(ka,km) = k\gcd(a,m)$.
negative_passages.docid: 102979
negative_passages.text: If $[G:H]$ and $[G:K]$ are relatively prime, then $G=HK$
negative_passages.docid: 81864
negative_passages.text: Finite group of order $mn$ with $\gcd(m,n) = 1$ .
negative_passages.docid: 53915
negative_passages.text: prove that $a \equiv b \mod m$ is an equivalence relation on the integers
negative_passages.docid: 21737
negative_passages.text: If $A|B$ and $B|A$ then prove $A=\pm B$
negative_passages.docid: 48607
negative_passages.text: If $A$ is normal and $A$ and $B$ commute, then $A^*$ and $B$ commute
negative_passages.docid: 47709
negative_passages.text: If $n\mid m$ prove that the canonical surjection $\pi: \mathbb Z_m \rightarrow \mathbb Z_n$ is also surjective on units
negative_passages.docid: 6410
negative_passages.text: Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law]
negative_passages.docid: 91947
negative_passages.text: Proof of $ f:I_n \rightarrow I_m \Rightarrow n = m$
negative_passages.docid: 24098
negative_passages.text: Prove that $a$ divides $b$ and $b$ divides $a$ if and only if $a = \pm b$