Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n
7091
If $\gcd(a,b)=1$, then $\gcd(a^n,b^n)=1$
139383
Prove that $\tau(2^n-1) \geq \tau(n)$ for all positive integers $n$.
139503
How to prove $|a+b|^k \leq 2^{k-1} (|a|^k+|b|^k)$?
29462
Suggestions on how to prove the following equality. $a^{m+n}=a^m a^n$
20878
If $(a,b)=1$ then prove $(a+b, ab)=1$.
130281
Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$
10236
Show that $a^n \mid b^n$ implies $a \mid b$
158844
If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $(a+1/a)^2+(b+1/b)^2\ge 25/2$
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
123275
Prove $[A,B^n] = nB^{n-1}[A,B]$
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
85977
Show that if $\sqrt{a}+\sqrt{b}\neq 0$, then $\mathbb{Q}(\sqrt{a}+\sqrt{b}) = \mathbb{Q}(\sqrt{a},\sqrt{b}), \forall a,b\in\mathbb{Q}$
89175
Proof of $(a+b)^{n+1}$
55428
Inductive proof of $a_n = 3*2^{n-1} + 2(-1)^n$ if $a_n = a_{n-1} + 2*a_{n-2}, a_1 = 1, a_2 = 8$
80601
Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
88995
Prove that $a^n-b^n = (a-b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$.
98855
Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$