query: Prove $\text{gcd}(a', b') = \text{gcd}(a, b)$
positive_passages.docid: 25825
positive_passages.text: Proof that $\gcd(ax+by,cx+dy)=\gcd(x,y)$ if $ad-bc= \pm 1$
negative_passages.docid: 128199
negative_passages.text: Proving gcd(a, a+2) is 1 or 2 for every integer a?
negative_passages.docid: 7091
negative_passages.text: If $\gcd(a,b)=1$, then $\gcd(a^n,b^n)=1$
negative_passages.docid: 113641
negative_passages.text: Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c)
negative_passages.docid: 19700
negative_passages.text: Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$
negative_passages.docid: 154959
negative_passages.text: Proof of associativity of GCD?
negative_passages.docid: 90480
negative_passages.text: Code for (a,b) with gcd(a,b)=1?
negative_passages.docid: 149993
negative_passages.text: Use the Euclidean algorithm to prove that gcd(na, nb) = n gcd(a, b).
negative_passages.docid: 101872
negative_passages.text: Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
negative_passages.docid: 157784
negative_passages.text: Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.
negative_passages.docid: 85688
negative_passages.text: How to show $\gcd(ac, bd) = \gcd(a,d) * \gcd(b,c)$
negative_passages.docid: 84417
negative_passages.text: Prove that if $\gcd(a,b)=1$ then $\gcd(a^m, b)=1$
negative_passages.docid: 34330
negative_passages.text: How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$?
negative_passages.docid: 69967
negative_passages.text: How to prove $\gcd(a^m-b^m,a^n-b^n) = a^{\gcd(m,n)} - b^{\gcd(m,n)} $?
negative_passages.docid: 147246
negative_passages.text: If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$
negative_passages.docid: 65218
negative_passages.text: Let $m \in \mathbb{Z^+} , n \in \mathbb{Z^+}$ and let $d=\gcd(m,n)$. Prove that $m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}$
negative_passages.docid: 78678
negative_passages.text: If $\gcd(m,n)=1$, and $\gcd(a,m)=\gcd(b,n)=1$, then $\gcd(ab, mn)=1$