Prove that if $n$ is an odd integer, then $n^2-1$ is divisible by $8$.
81601
Prove that if $x$ is odd then $x^2 -1$ is divisible by $8$.
12729
Proving $9^n - 8n - 1$ is divisible by $8^2$ for $n\ge 0$?
91921
Divisibility of $n\cdot2^n+1$ by 3.
13525
Prove: If $n=2^k-1$, then $\binom{n}{i}$ is odd for $0\leq i\leq n$
156641
Show that $\gcd(3n,3n+ 2) = 1$ when $n$ is odd
72511
If $n$ is an odd natural number, then $8$ divides $n^{2}-1$
26514
Prove that if $n$ is a composite, then $2^n-1$ is composite.
17047
Prove that if n is even then $n^2$ is even and if n is odd then $n^2$ is odd?
6165
Proof by Induction $n^2+n$ is even
45037
Show $(2^m-1,2^n+1)=1$ if $m$ is odd
31890
Prove using induction that from a set of $n+1$ numbers from $\{1,2,..2n\}$, at least one number will evenly divide another.
3611
How to prove $n^5 - n$ is divisible by $30$ without reduction
101927
Induction: Prove that $4^{n+1}+5^{2 n - 1}$ is divisible by 21 for all $n \geq 1$.
7414
Prove that odd perfect square is congruent to $1$ modulo $8$
85817
Show that in any set of $2n$ integers, there is a subset of $n$ integers whose sum is divisible by $n$.
1837
Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
94983
Can an odd number $n$ divide $2^n-1$?