Prove that $(x^n - 1)$ can divide $(x^{kn} - 1)$ without any remainder.
62732
Prove：$x^d-1 \mid x^n-1$ iff $d \mid n$.
142534
Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
26336
Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
156766
Prove that $\sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$
116363
If $P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x)$ , then prove that $P(x)$ is divisible by $x-1$
99052
Prove that $( 1 + n^{-2}) ^n \to 1$.
55266
Prove that $(n+1)^{n-1}<n^n$
31890
Prove using induction that from a set of $n+1$ numbers from $\{1,2,..2n\}$, at least one number will evenly divide another.
115866
Proof by induction; $a^n$ divides $b^n$ implies $a$ divides $b$
97469
How to prove $|a^k|=n/\gcd(n,k)$ whenever $|a|=n$?
34154
Prove the statement $\forall x\in\mathbb N (x > 1\to\exists k\in\mathbb N\exists m \in\mathbb N (m \equiv 1 (\text{mod }2) \wedge x = 2^km))$.
136737
Proof by Induction for inequality, $\sum_{k=1}^nk^{-2}\lt2-(1/n)$
138083
$\gcd(X^n - 1, nX^{n-1}) = 1$ in a field $K$ of characteristic $0$ or characteristic $p \nmid n$.
53866
Prove $ x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$
31555
Divisor of $m$ and $n$ divides $m - qn$ (in proof of Euclidean algorithm)
83752
If $p + q = 1$ prove that for any natural $n, m$ following is true: $(1 - p^n)^m + (1 - q^m)^n \ge 1$
124131
Show that $(x-1)^2$ is a factor of $x^n -nx +n-1$