Show that $24 \mid (n - 1)n(n + 1)(n + 2)$ for all positive integers $n$.
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 $24$
1837
Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
35342
Show that $ \sum_{n=2}^m \binom{n}{2} = \binom{m+1}{3}$
62556
Prove $3|n(n+1)(n+2)$ by induction
119118
Prove $n\mid \phi(2^n-1)$
130488
Prove by induction that $3\mid n^3 - n$
83407
How to show that that $q \mid p-1$?
42526
The product of $n$ consecutive integers is divisible by $ n!$ (without using the properties of binomial coefficients)
141088
Proof : $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$.
123155
If $mn+1$ is a multiple of $24$, prove that $m+n$ is also a multiple of 24.
87522
If $ a + b + c \mid a^2 + b^2 + c^2$ then $ a + b + c \mid a^n + b^n + c^n$ for infinitely many $n$
136459
Prove ${n - 1 \choose k - 1} + {n - 2 \choose k - 1} + {n - 3 \choose k - 1} + \dots + {k - 1 \choose k - 1} = {n \choose k}$
53857
Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$
6165
Proof by Induction $n^2+n$ is even
959
Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
99634
$\binom{n}{1},\binom{n}{2},\ldots ,\binom{n}{n-1}$ are all even numbers.
50894
Prove that $6$ divides $n(n + 1)(n + 2)$