If $a$ is an odd integer, then prove that $24\mid a(a^2 - 1)$
118574
prove that $24 \mid a(a^2-1)$
74561
Let $a_1,a_2,\cdots,a_n$ be $n$ numbers such that $a_i$ is either $1$ or $-1$.If $a_1a_2a_3a_4+\cdots+a_na_1a_2a_3=0$ then prove that $4\mid n$.
513
If n is an odd integer, show there exists a positive integer k such that 2^k mod n = 1.
139230
Prove that for every positive integer, this polynomial is divisible by 24.
119118
Prove $n\mid \phi(2^n-1)$
32803
Prove that if b is coprime to 6 then $b^2 \equiv 1 $ (mod 24)
157294
Prove that if $n>1$ is odd, then $1^n+2^n+3^n+\ldots+(n-1)^n$ is divisible by $n$
136304
Prove: $2^b-1\mid{}2^a-1 \iff b\mid{}a$
97666
Prove $133\mid \left(11^{n+2}+12^{2n+1}\right)$
17047
Prove that if n is even then $n^2$ is even and if n is odd then $n^2$ is odd?
114873
Show that if $n$ is not divisible by $2$ or by $3$, then $n^2-1$ is divisible by $24$.
22093
How to show $a^{2^n}+1 \mid a^{2^m}-1$?
118629
$p,q$ odd primes, $q\mid a^p-1\Rightarrow q\mid a-1$ or $\,2p\mid q-1$
12100
$n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $n \mid \frac{a^{n}-b^{n}}{a-b}$
130027
Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z$
50410
If $A=\pmatrix{1 &0\\-1&1}$, show that $A^2-2A+I_2=0$. Hence find $A^{50}$
90916
Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.