For any integer $n$, $n^2$ is congruent to $0$ or $1$ modulo $3$
135212
The square of an integer has form $3n$ or $3n+1$
115435
Induction: prove using congruences
74512
How to show $n(n+1)(2n+1) \equiv 0 \pmod 6$?
76032
Prove that if m is a square integer then m is neither congruent to 2 modulo 5 nor congruent to 3 modulo 5
112390
Solutions of $x^2 \equiv 1 \pmod N$
75166
Find with proof all positive integers $n$ such that $2^n + 1 \equiv 0 \pmod{n}$
77383
Prove that $n^3=n \text{ mod }6$ for every integer $n$.
513
If n is an odd integer, show there exists a positive integer k such that 2^k mod n = 1.
142008
Find all natural numbers $n$ such that ${(n+1)}^{1/2}-{(n-1)}^{1/2}$ is rational
127816
Number theory proof mod p
29640
If $x \equiv 1 \pmod 3$ and $x \equiv 0 \pmod 2$, what is $x \pmod 6$?
95653
Euler's theorem: [3]^2014^2014 mod 98
108238
Find the smallest integer $n$ greater than $1$ such that the last $3$ digits of $n^2$ are the same as the last $3$ digits of $n$.
6160
How to prove a number $a$ is quadratic residue modulo $n$?
127901
If $x^2+y^2 \equiv 0\pmod{p}$, then $p \equiv 1 \pmod{4}$
10191
Binomial theorem: divisibility by $n^2$
122914
An integer $n$, such that $nx = 0$, where $x$ belongs to the quotient group $\Bbb Q/\Bbb Z$