Show that if $x^d -1$ divides $x^n -1$ if and only if $d$ divides $n$
62732
Prove：$x^d-1 \mid x^n-1$ iff $d \mid n$.
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
49687
Show that the number 9 divides the number $ m$ if and only if the sum of the digits of the number $ m $ is divisible by 9.
19599
$ d(x_n,y_n) \to d(x,y)$ if $x_n\to x$ and $y_n \to y$
36906
Prove that $x^n+x^{-n} \in \mathbf{N}$ if $x+\frac1x \in \mathbf{N}$
117167
$3^{n+1}$ divides $2^{3^n}+1$
24098
Prove that $a$ divides $b$ and $b$ divides $a$ if and only if $a = \pm b$
124131
Show that $(x-1)^2$ is a factor of $x^n -nx +n-1$
156356
Proof that $x^n+\frac{1}{x^n}\in \mathbb Z$ by complete induction
33996
Proving that $\gcd(2^m - 1, 2^n - 1) = 2^{\gcd(m,n )} - 1$
136854
Prove: For a,b,c positive integers, ac divides bc if and only if a divides b
153313
Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
103854
Showing that $Q_n=D_n+D_{n-1}$
96152
Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
55266
Prove that $(n+1)^{n-1}<n^n$
7090
Show that $\gcd(a,bc)=1$ if and only if $\gcd(a,b)=1$ and $\gcd(a,c)=1$
66145
>Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$.