Prove that if $a$ is odd, then $a^2\equiv 1\pmod 8$
74523
Proof by Cases involving divisibility
13525
Prove: If $n=2^k-1$, then $\binom{n}{i}$ is odd for $0\leq i\leq n$
102234
Show $a^p \equiv a \mod 2p$
81601
Prove that if $x$ is odd then $x^2 -1$ is divisible by $8$.
5708
Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
152299
Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
75166
Find with proof all positive integers $n$ such that $2^n + 1 \equiv 0 \pmod{n}$
31064
Showing $x^8\equiv 16 \pmod{p}$ is solvable for all primes $p$
17601
How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$
49548
If $(a + ib)^3 = 8$, prove that $a^2 + b^2 = 4$
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
153532
Proving $a^2$ is even $\implies$ $a$ is even by contradiction
139
Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)
24466
Prove that if $[F(\alpha):F]$ is odd then $F(\alpha)=F(\alpha^2)$
94570
Proof that every odd integer is a difference of two squares
32803
Prove that if b is coprime to 6 then $b^2 \equiv 1 $ (mod 24)
94326
a square in a finite field of odd order