Show that $n(n^2-1)$ is divisible by 24, if $n$ is an odd integer greater than $2$.
17965
Prove 24 divides $u^3-u$ for all odd natural numbers $u$
42526
The product of $n$ consecutive integers is divisible by $ n!$ (without using the properties of binomial coefficients)
108382
Prove $x^n-1$ is divisible by $x-1$ by induction
2554
Show that if $n>2$, then $(n!)^2>n^n$.
99052
Prove that $( 1 + n^{-2}) ^n \to 1$.
70386
Proof by induction: $2^n > n^2$ for all integer $n$ greater than $4$
124816
Using the fact that $n^3 + 5n$ is divisible by $6$, infer that $n^4 - 2n^3 + 11n^2 + 14n$ is always divisible by $24$
72511
If $n$ is an odd natural number, then $8$ divides $n^{2}-1$
26336
Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
94983
Can an odd number $n$ divide $2^n-1$?
24053
How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$?
72054
Show that $\deg f$ is even when $n$ is odd.
129238
Divisibility of $2^n - 1$ by $2^{m+n} - 3^m$.
156641
Show that $\gcd(3n,3n+ 2) = 1$ when $n$ is odd
122711
Prove by induction that $n^3 + 11n$ is divisible by $6$ for every positive integer $n$.
68328
Infinitely many $n$ such that $2^n-n$ is divisible by a prime
97805
$n+1$ is a divisor of $\binom{2n} {n}$