query: $\mathrm{gcd}(a+nb,b)=\mathrm{gcd}(a,b)$ for all $n\in\mathbb{Z}$
positive_passages.docid: 102133
positive_passages.text: Prove: $\gcd(a,b) = \gcd(a, b + at)$.
negative_passages.docid: 21788
negative_passages.text: Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$
negative_passages.docid: 130112
negative_passages.text: If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
negative_passages.docid: 147246
negative_passages.text: If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$
negative_passages.docid: 11600
negative_passages.text: $\gcd(a,\operatorname{lcm}(b,c))=\operatorname{lcm}(\gcd(a,b),\gcd(a,c))$
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: 77278
negative_passages.text: If $\,\gcd(a,b) = {\rm lcm}(a,b)\,$ then prove $a=b$
negative_passages.docid: 85688
negative_passages.text: How to show $\gcd(ac, bd) = \gcd(a,d) * \gcd(b,c)$
negative_passages.docid: 66748
negative_passages.text: Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
negative_passages.docid: 38295
negative_passages.text: If $\gcd (a,b) = 1$, what can be said about $\gcd (a+b,a-b)$?
negative_passages.docid: 77064
negative_passages.text: Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$
negative_passages.docid: 55441
negative_passages.text: Check: For the integers $a,b,c$ show that $\gcd(a,bc)=\gcd(a,\gcd(a,b)\cdot c)$
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: 25573
negative_passages.text: If $\gcd(a,b)=1$, then $\gcd(a+b,a^2 -ab+b^2)=1$ or $3$.
negative_passages.docid: 1713
negative_passages.text: number theory $\gcd(a,bc)=\gcd(a,c)$
negative_passages.docid: 103147
negative_passages.text: $\operatorname{gcd}(ab,a+b)=1$ if $a$ and $b$ are relatively prime
negative_passages.docid: 25907
negative_passages.text: Show $a\Bbb Z+b\Bbb Z = \gcd(a,b)\Bbb Z$