Determine necessary and sufficient conditions on $x, N$ so that the following holds: for any $a, b$, if $ax ≡ bx \pmod N$, then $a ≡ b\pmod N$
54430
For $a,b$ coprime, there exists positive integers $x,y$ such that $ax-by=1$
75199
$a \equiv b \pmod n$ and $c\equiv d \pmod n$ implies $ac \equiv bd \pmod n$
29640
If $x \equiv 1 \pmod 3$ and $x \equiv 0 \pmod 2$, what is $x \pmod 6$?
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
157924
$\forall x\in\mathbb{N}[(\exists y\in\mathbb{N}(2\leq y\land y<x\land y\ |\ x))\to(\exists z\in\mathbb{N}(2\leq z\land z\leq\sqrt x\land z\ |\ x))]$
151038
How to prove (A v B), (A → C), (B → D) therefore (C v D)
21781
Prove for any $a>1$ there exist a natural number, $N$, such that $a^n>n$ for all $n \ge N$
142360
How to show $x_1,x_2, \dots ,x_n \geq 0 $ and $ x_1 + x_2 + \dots + x_n \leq \frac{1}{2} \implies (1-x_1)(1-x_2) \cdots (1-x_n) \geq \frac{1}{2}$
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
32155
Prove that if for every $x \in \mathbb{R}^N$ $Ax=Bx$ then $A=B$
64955
Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$
83996
For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$
29204
$a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus
92290
For each $x,y,z∈\mathbb N$, if $x<y$ then $x+z<y+z$
978
Solving $ax \equiv c \pmod b$ efficiently when $a,b$ are not coprime
132587
Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$
45939
For which $n≥2$ does there exist $\sigma\in A_n$ with $|\sigma| > n$