query: Smallest positive element of $ \{ax + by: x,y \in \mathbb{Z}\}$ is $\gcd(a,b)$
positive_passages.docid: 19656
positive_passages.text: Proving that $ \gcd(a,b) = as + bt $, i.e., $ \gcd $ is a linear combination.
negative_passages.docid: 27544
negative_passages.text: With $a,\ b,\ m,\ n\in \mathbb{Z}$. Show that if ${\rm gcd}(m,n)=1$, then ${\rm gcd}(ma+nb,mn)={\rm gcd}(a,n){\rm gcd}(b,m)$
negative_passages.docid: 19500
negative_passages.text: How can I find $\gcd(a^m+1,a^n+1)$ with $a,m,n$ positive integers?
negative_passages.docid: 94770
negative_passages.text: Greatest common divisor of two positive integers
negative_passages.docid: 29608
negative_passages.text: If GCD $(a_1,\ldots, a_n)=1$ then there's a matrix in $SL_n(\mathbb{Z})$ with first row $(a_1,\ldots, a_n)$
negative_passages.docid: 103147
negative_passages.text: $\operatorname{gcd}(ab,a+b)=1$ if $a$ and $b$ are relatively prime
negative_passages.docid: 157784
negative_passages.text: Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.
negative_passages.docid: 7316
negative_passages.text: How to prove that $z\gcd(a,b)=\gcd(za,zb)$
negative_passages.docid: 35945
negative_passages.text: Proof that if $\gcd(a,b) = 1$ and $a\mid n$ and $b\mid n$, $ab \mid n$
negative_passages.docid: 147079
negative_passages.text: $\gcd\left(\binom{n}{k},n\right)>1$ for nontrivial values?
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: 137626
negative_passages.text: If $a,b,c$ are positive integers,and $\cfrac{a}{b}+\cfrac{b}{c}+\cfrac{c}{a}$ and $\cfrac{b}{a}+\cfrac{c}{b}+\cfrac{a}{c}$ are integers then $a=b=c$
negative_passages.docid: 2208
negative_passages.text: $\gcd(a,b,c)=\gcd(\gcd(a,b),c)\,$ [GCD Associative Law]
negative_passages.docid: 84491
negative_passages.text: Finding the smallest positive integer $n$ such that $S_n$ contains an element of order 60.
negative_passages.docid: 117098
negative_passages.text: Properties of the greatest common divisor: $\gcd(a, b) = \gcd(a, b-a)$ and $\gcd(a, b) = \gcd(a, b \text{ mod } a)$
negative_passages.docid: 115850
negative_passages.text: Minimum of $|ax-by+c|$
negative_passages.docid: 73716
negative_passages.text: Let $x,y,z>0$ and $x+y+z=1$, then find the least value of ${{x}\over {2-x}}+{{y}\over {2-y}}+{{z}\over {2-z}}$