query: Greatest common divisor of $X^n-1$ and $X^m-1$
positive_passages.docid: 6012
positive_passages.text: Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
negative_passages.docid: 36862
negative_passages.text: A question about proving that there is no greatest common divisor
negative_passages.docid: 132587
negative_passages.text: Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$
negative_passages.docid: 19500
negative_passages.text: How can I find $\gcd(a^m+1,a^n+1)$ with $a,m,n$ positive integers?
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: 72640
negative_passages.text: $(m,n) = (\gcd(m,n))$ as ideals in $\Bbb Z$
negative_passages.docid: 63769
negative_passages.text: Group theory proof of Euler's theorem ($a^{\phi(m)} \equiv 1\mbox{ }(\mbox{mod }m)$ if $\gcd(a,m)=1$)
negative_passages.docid: 124279
negative_passages.text: $g^k$ is a primitive element modulo $m$ iff $\gcd (k,\varphi(m))=1$
negative_passages.docid: 52483
negative_passages.text: Showing that $(m)\cap (n)=(\operatorname{lcm}(m,n))$ and $(m)+(n)=(\gcd(m,n))$ for any $m,n\in\mathbb{Z}$
negative_passages.docid: 136077
negative_passages.text: Prove that $x-1$ is a factor of $x^n-1$
negative_passages.docid: 97805
negative_passages.text: $n+1$ is a divisor of $\binom{2n} {n}$
negative_passages.docid: 117167
negative_passages.text: $3^{n+1}$ divides $2^{3^n}+1$
negative_passages.docid: 48452
negative_passages.text: Proof that $\gcd$ divides $\operatorname{lcm}$
negative_passages.docid: 8602
negative_passages.text: Galois Group of $(x^2-p_1)\cdots(x^2-p_n)$
negative_passages.docid: 5708
negative_passages.text: Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
negative_passages.docid: 152506
negative_passages.text: The greatest common divisor is the smallest positive linear combination
negative_passages.docid: 65218
negative_passages.text: Let $m \in \mathbb{Z^+} , n \in \mathbb{Z^+}$ and let $d=\gcd(m,n)$. Prove that $m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}$