Proving that $x^5 = x \pmod{10}$ for every integer $x$.
73307
Why is the last digit of $n^5$ equal to the last digit of $n$?
155743
Why does the equation $x^2\equiv2 \pmod 5$ have no solutions?
7085
Prove that if the equation $x^{2} \equiv a\pmod{pq}$ has any solutions, then it has four solutions.
76032
Prove that if m is a square integer then m is neither congruent to 2 modulo 5 nor congruent to 3 modulo 5
146053
Proving identities (mod $pq$) using Fermat's little theorem?
21874
Solve $x^3+x \equiv 1 \pmod p$
71788
Prove existence of multiplicative inverse modulo p when p is prime
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
59566
Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
31064
Showing $x^8\equiv 16 \pmod{p}$ is solvable for all primes $p$
103545
Prove "casting out nines" of an integer is equivalent to that integer modulo 9
112575
Finding integer solutions of $m^2-n^5 = m - n$
63416
Prove $(n^5-n)$ is divisible by 5 by induction.
49299
Proof by Induction: For any integer $n > 0$, $3|(5^{2n} − 1)$.
120227
Proving $6^n - 1$ is always divisible by $5$ by induction
3611
How to prove $n^5 - n$ is divisible by $30$ without reduction
94017
If p is a prime and a ∈Z. How do we prove that the congruence x^x ≡ a (mod p) is solvable?