If $(a,b)=1$ then there exist positive integers $x$ and $y$ s.t $ax+by=1$.
11388
How you get from $\gcd(a,b)=1 \Longleftrightarrow ax+by=1 \Longrightarrow a \nmid b$?
20878
If $(a,b)=1$ then prove $(a+b, ab)=1$.
422
If $\gcd(a,b)=1$, $\gcd(a,y)=1$ and $\gcd(b,x)=1$ then prove that $ax+by$ is prime to $ab$
6970
If $abc=1$ and $a,b,c$ are positive real numbers, prove that ${1 \over a+b+1} + {1 \over b+c+1} + {1 \over c+a+1} \le 1$.
56818
Prove that if $A$ is positive, then $\exists\;\alpha>0$ such that $\langle Ax, x \rangle\geq \alpha|x|^2$ for $x\in\Bbb{R}^n$
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
121700
If $a$ and $b$ are positive real numbers, then $a + b \geq 2 \sqrt{ab}$.
92290
For each $x,y,z∈\mathbb N$, if $x<y$ then $x+z<y+z$
127183
Proof: For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$
124320
$a,b,c>0 \text{ s.t. }a+b+c=1 \implies \sqrt{ab+c}+\sqrt{bc+a}+\sqrt{ca+b} \ge 1+ \sqrt{ab}+\sqrt{bc}+\sqrt{ca}$
20593
Show that if $a \neq b$ and a and b are positive then $\frac{a}{b}+\frac{b}{a}$ is never an integer
45090
Given any positive real numbers $a,b,c$, we have $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
114404
For which positive integer values of $n$ does $a|mn-b?$
64654
Suppose $a, b$ and $n$ are positive integers. Prove that $a^n$ divides $b^n$ if and only if $a$ divides $b$.
12096
If $a+1/a$ is an integer, then so is $a^t+1/a^t$ for $t\in\mathbb N$
135750
Prove that if a and b are integers, then there are unique integers q and r such that $a = bq + r$, $-|b|/2 < r \le |b|/2$
35874
Show that when $BA = I$, the solution of $Ax=b$ is unique