query: How to show that greatest common factor of $(m+n$ and $ m-n)\in\mathbb{N}$ is equal to $1$ or $2$?
positive_passages.docid: 38404
positive_passages.text: Prove $\gcd(a+b, a-b) = 1$ or $2\,$ if $\,\gcd(a,b) = 1$
negative_passages.docid: 67660
negative_passages.text: Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
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: 105832
negative_passages.text: prove that if $\gcd(a,b)=1$ then $\gcd(a^2, b^2)=1$
negative_passages.docid: 83307
negative_passages.text: If $n\mid a^2-1$, $n$ an odd integer, show $n = \gcd(a-1, n)\gcd(a+1, n)$
negative_passages.docid: 2289
negative_passages.text: Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
negative_passages.docid: 59566
negative_passages.text: Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
negative_passages.docid: 34738
negative_passages.text: Prove that $\gcd(n!+1,(n+1)!+1)=1$
negative_passages.docid: 24845
negative_passages.text: Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
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: 37808
negative_passages.text: Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$
negative_passages.docid: 88124
negative_passages.text: $M,N\in \Bbb R ^{n\times n}$, show that $e^{(M+N)} = e^{M}e^N$ given $MN=NM$
negative_passages.docid: 66748
negative_passages.text: Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
negative_passages.docid: 137739
negative_passages.text: Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$
negative_passages.docid: 59006
negative_passages.text: Proof of $\gcd(n!+1, (n+1)!+1)=1$?
negative_passages.docid: 117098
negative_passages.text: Properties of the greatest common divisor: $\gcd(a, b) = \gcd(a, b-a)$ and $\gcd(a, b) = \gcd(a, b \text{ mod } a)$
negative_passages.docid: 30752
negative_passages.text: Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$