query: For positive integers $𝑛$, $𝑎$ and $𝑏$ with $𝑎𝑏 = 𝑛^2$ and $\gcd(a,b)=1$, show that $a$ and $b$ are squares
positive_passages.docid: 155784
positive_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: 19656
negative_passages.text: Proving that $ \gcd(a,b) = as + bt $, i.e., $ \gcd $ is a linear combination.
negative_passages.docid: 101872
negative_passages.text: Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
negative_passages.docid: 128199
negative_passages.text: Proving gcd(a, a+2) is 1 or 2 for every integer a?
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: 75039
negative_passages.text: Smallest positive element of $ \{ax + by: x,y \in \mathbb{Z}\}$ is $\gcd(a,b)$
negative_passages.docid: 2289
negative_passages.text: Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
negative_passages.docid: 108450
negative_passages.text: Prove that $\gcd(a, b) = \gcd(a, b + ma)$?
negative_passages.docid: 34738
negative_passages.text: Prove that $\gcd(n!+1,(n+1)!+1)=1$
negative_passages.docid: 92630
negative_passages.text: How to prove $\gcd(a^m,b^m) = \gcd(a,b)^m$ using Bézout's Lemma
negative_passages.docid: 37808
negative_passages.text: Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$
negative_passages.docid: 126358
negative_passages.text: Prove for $\gcd$ of two integers $a,b$, that $(a^n,b^n) = (a,b)^n$
negative_passages.docid: 10348
negative_passages.text: Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
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: 102154
negative_passages.text: If $a$ and $b$ are consecutive integers, prove that $a^2 + b^2 + a^2b^2$ is a perfect square.
negative_passages.docid: 7316
negative_passages.text: How to prove that $z\gcd(a,b)=\gcd(za,zb)$
negative_passages.docid: 22845
negative_passages.text: Prove that if d is a common divisor of a and b, then $d=\gcd(a,b)$ if and only if $\gcd(a/d,b/d)=1$