Given integers $a$ and $b$ prove that if there exist integers $x$ and $y$ for which $ax + by = (a, b)$, then $(x, y) = 1$.
65687
$\gcd(a,b)\gcd(x,y)\mid ax+by$
31937
How do I prove equality of x and y?
149287
Prove that $A,B>0$ and $A_{ij}0$
31436
Show that for all $a\in\mathbb{N}$, there exists $b\in\mathbb{N}$ and square-free integer $c$ such that $\sqrt{a}=b\sqrt{c}$.
154327
$A^2X = X$ show that $AX = X$, if $A_{ij} > 0$ and $x_{i} > 0$
110408
Solving for $A$ in $Ax = b$
124320
$a,b,c>0 \text{ s.t. }a+b+c=1 \implies \sqrt{ab+c}+\sqrt{bc+a}+\sqrt{ca+b} \ge 1+ \sqrt{ab}+\sqrt{bc}+\sqrt{ca}$
127183
Proof: For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$
41364
How to prove that $x^y - y^x = x + y$ has only one solution in positive integers?
142724
Prove that for all integers $x$ and $y$, $x - y$ is odd if and only if $x + y$ is odd.
25784
Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$.
25825
Proof that $\gcd(ax+by,cx+dy)=\gcd(x,y)$ if $ad-bc= \pm 1$
16774
If $x,y,z\in\mathbb R\setminus \{1\}$ and $xyz=1$, prove that $\frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1$.
47856
$f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$
64955
Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$
63535
Prove that if $A – C = B – C$ and $A\cap C = B\cap C$ then $A = B$
71012
If $a, b$ are positive integers such that $a|b^2, b^2|a^3,....$ then $a=b$.