query: Prove $\gcd(a^n,b^n) = 1 \Rightarrow \gcd(a,b)=1$
positive_passages.docid: 126358
positive_passages.text: Prove for $\gcd$ of two integers $a,b$, that $(a^n,b^n) = (a,b)^n$
negative_passages.docid: 72424
negative_passages.text: Suppose 1 ≤ k ≤ n − 1 and gcd(k, n) = 1. Prove that gcd(n − k, n) = 1
negative_passages.docid: 82938
negative_passages.text: $\gcd(a,b)=1, x^a = y^b\Rightarrow x = n^b$, $ y = n^a$ for an integer $n$.
negative_passages.docid: 112031
negative_passages.text: Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers.
negative_passages.docid: 42780
negative_passages.text: Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$.
negative_passages.docid: 51858
negative_passages.text: Showing $\gcd(2^m-1,2^n+1)=1$
negative_passages.docid: 7151
negative_passages.text: Let $a,m,n \in \mathbf{N}$. Show that if $\gcd(m,n)=1$, then $\gcd(a,mn)=\gcd(a,m)\cdot\gcd(a,n)$.
negative_passages.docid: 66785
negative_passages.text: Find $\gcd(p^n-1,p^m+1)$.
negative_passages.docid: 81116
negative_passages.text: $gh = hg, \ \gcd(|g|, |h|) = 1\Rightarrow|gh| = |g||h|\ \ (|a|$ = order of element $a)$
negative_passages.docid: 95889
negative_passages.text: Prove that $\gcd(n! + 1, 3n! + 1) = 1$, $\forall n \in \mathbb{N}$, $n \geq 2$.
negative_passages.docid: 18714
negative_passages.text: $\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1$, or $p$
negative_passages.docid: 74631
negative_passages.text: For natural numbers $a$ and $b$, show that $a \Bbb Z + b \Bbb Z = \gcd(a, b)\Bbb Z $
negative_passages.docid: 88995
negative_passages.text: Prove that $a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$.
negative_passages.docid: 1713
negative_passages.text: number theory $\gcd(a,bc)=\gcd(a,c)$
negative_passages.docid: 102133
negative_passages.text: Prove: $\gcd(a,b) = \gcd(a, b + at)$.
negative_passages.docid: 158308
negative_passages.text: Proving that if $a,b$ are even, then $\gcd(a,b) = 2 \gcd(a/2, b/2)$
negative_passages.docid: 20477
negative_passages.text: Prove that $\gcd(3^n-2,2^n-3)=\gcd(5,2^n-3)$