Prove that for any integers $x, y$ and $z$ and any integer $k$that if $x \equiv_k y$ and $y \equiv_k z$ then $x \equiv_k z$
11342
Show that conguence $\bmod m\,$ is an equivalence relation
86285
Given positive integers $ x , y$ and that $3x^2 + x = 4y^2 +y$ Prove that $x-y$ is a perfect square
92290
For each $x,y,z∈\mathbb N$, if $x<y$ then $x+z<y+z$
142534
Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
121853
$X$,$Y$,$Z$ mutually independent implies $X+Y$ independent of $Z$
70951
Prove divisibility law $\,b\mid a,c\,\Rightarrow\, b\mid ka + lc$ for all $k,l\in\Bbb Z$
34693
Prove $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$ if $E^{\mathbb P}[X]=\mathbb P(X>0)=1$ and $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $
26301
Prove that $x^3 \equiv x \bmod 6$ for all integers $x$
149287
Prove that $A,B>0$ and $A_{ij}0$
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
36
Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$
102175
$Z=XY$, $E_Z[Z]$, $E_{X,Y}[XY]$
137626
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 $a=b=c$
143543
Prove that $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$
121717
Prove that $z^2 \equiv ab\pmod p$ is solvable if and only if both or neither of $x^2 \equiv a \pmod p$, $x^2 \equiv b \pmod p$ are solvable.
83738
Prove $(k,n+k)=1$ iff $(k,n)=1$
7316
How to prove that $z\gcd(a,b)=\gcd(za,zb)$