How to prove that $42|a^7-a$?
24053
How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$?
64782
How to prove if $n$ is prime and $n | a^2$ then $n | a$?
100892
Proving divisibility of $a^3 - a$ by $6$
43913
Prove that if $A^*+A=AA^*$ then $A$ is normal
83662
Prove that $ a $ divides $1^{a^{n}} + 2^{a^{n}} + ... + (a-1)^{a^{n}}$?
96535
Show that if $\gcd(a,3)=1$ then $a^7 \equiv a\pmod{63}$. Why is this assumption necessary?
69967
How to prove $\gcd(a^m-b^m,a^n-b^n) = a^{\gcd(m,n)} - b^{\gcd(m,n)} $?
90962
For positive integers $m,n$ if $\sqrt 7 - \frac{m}{n} > 0$ then prove that $\sqrt 7 - \frac{m}{n} > \frac{1}{{mn}}$
21781
Prove for any $a>1$ there exist a natural number, $N$, such that $a^n>n$ for all $n \ge N$
50614
If $4\mid a+bc$ and $6\mid b+ac$ prove that $2\mid a^2-b^2$
158807
Proving a fact: $\tan(6^{\circ}) \tan(42^{\circ})= \tan(12^{\circ})\tan(24^{\circ})$
96152
Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
55266
Prove that $(n+1)^{n-1}<n^n$
52055
how to prove $437\,$ divides $18!+1$? (NBHM 2012)
153532
Proving $a^2$ is even $\implies$ $a$ is even by contradiction
145041
Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
98855
Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$