query: If $\gcd(p,q+Ar)>1$ for all $A\in \mathbb{Z}$, then $\gcd(p,q,r)>1$
positive_passages.docid: 36
positive_passages.text: Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$
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: 4141
negative_passages.text: Prove: If $a\mid m$ and $b\mid m$ and $\gcd(a,b)=1$ then $ab\mid m$
negative_passages.docid: 7316
negative_passages.text: How to prove that $z\gcd(a,b)=\gcd(za,zb)$
negative_passages.docid: 128820
negative_passages.text: Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$
negative_passages.docid: 190
negative_passages.text: Why $\gcd(qb+r,b)=\gcd(b,r)$?
negative_passages.docid: 73391
negative_passages.text: Prove that if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$
negative_passages.docid: 102133
negative_passages.text: Prove: $\gcd(a,b) = \gcd(a, b + at)$.
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: 59006
negative_passages.text: Proof of $\gcd(n!+1, (n+1)!+1)=1$?
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: 78678
negative_passages.text: If $\gcd(m,n)=1$, and $\gcd(a,m)=\gcd(b,n)=1$, then $\gcd(ab, mn)=1$
negative_passages.docid: 155784
negative_passages.text: If $\gcd(a, b) = 1$ and if $ab = x^2$, prove that $a, b$ must also be perfect squares; where $a,b,x$ are in the set of natural numbers
negative_passages.docid: 97469
negative_passages.text: How to prove $|a^k|=n/\gcd(n,k)$ whenever $|a|=n$?
negative_passages.docid: 31331
negative_passages.text: $|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
negative_passages.docid: 66785
negative_passages.text: Find $\gcd(p^n-1,p^m+1)$.
negative_passages.docid: 72424
negative_passages.text: Suppose 1 ≤ k ≤ n − 1 and gcd(k, n) = 1. Prove that gcd(n − k, n) = 1