query: Prove that if both $x$ and $y$ are odd, then $\gcd(x, y)=\gcd\left(\frac{x-y}{2}, y\right)$
positive_passages.docid: 24160
positive_passages.text: Show that if $a$ is an odd integer and $b$ is an even integer then $\gcd(a,b)=\gcd(a,b/2)$
negative_passages.docid: 20477
negative_passages.text: Prove that $\gcd(3^n-2,2^n-3)=\gcd(5,2^n-3)$
negative_passages.docid: 97411
negative_passages.text: If $x^n=y^n$ and $n$ is odd then $x=y$
negative_passages.docid: 31465
negative_passages.text: Prove that if $a \mid b$ then $\gcd(a,b) = |a|$
negative_passages.docid: 76507
negative_passages.text: $\gcd(b^x - 1, b^y - 1, b^ z- 1,\dots) = b^{\gcd(x, y, z,\dots)} -1$
negative_passages.docid: 34738
negative_passages.text: Prove that $\gcd(n!+1,(n+1)!+1)=1$
negative_passages.docid: 1713
negative_passages.text: number theory $\gcd(a,bc)=\gcd(a,c)$
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: 23066
negative_passages.text: If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$
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: 7316
negative_passages.text: How to prove that $z\gcd(a,b)=\gcd(za,zb)$
negative_passages.docid: 37808
negative_passages.text: Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$
negative_passages.docid: 66748
negative_passages.text: Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
negative_passages.docid: 76680
negative_passages.text: Prove that if $\gcd(a,c)=1$ and $\gcd(b,c)=1$ then $\gcd(ab,c)=1$
negative_passages.docid: 136064
negative_passages.text: How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?
negative_passages.docid: 20003
negative_passages.text: Prove for positive integers a,b,c and d (where b does not equal d), if gcd(a,b) = gcd(c,d) = 1, then a/b + c/d is not an integer
negative_passages.docid: 108450
negative_passages.text: Prove that $\gcd(a, b) = \gcd(a, b + ma)$?