Proving that if $a^n \equiv 1 \pmod m$ then $d\mid n$
4972
Mod of numbers with large exponents [modular order reduction]
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
22093
How to show $a^{2^n}+1 \mid a^{2^m}-1$?
35948
How to prove $D^n/S^{n-1}\cong S^n$?
62996
A problem regarding the proof of ${p^nk\choose p^n}\equiv k\mod p$, where $p\nmid k$.
78356
How to prove ${n \choose p} \equiv\left\lfloor \frac{n}{p} \right\rfloor\pmod p$
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
91935
If $x \equiv a \pmod {p_1}$ and $x\equiv a \pmod{p_2}$, then is it true that $x\equiv a \pmod{p_1p_2} ?$
138894
Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
141088
Proof : $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$.
112631
Prove that if $d$ divides $n$, then $2^d -1$ divides $2^n -1$
145041
Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
39217
If $d \mid n$ then $x^{q^d} - x \mid x^{q^n} - x$.
11989
$ac\equiv bc \pmod{\!m}\!\!\iff\!\! a\equiv b \pmod{\!\!\frac{m}{(c,m)}}$ [Euclid's Lemma, congruence form]
1712
$ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property]
44033
Number theory lemma - $x^2 \equiv 1 \pmod p$ implies $x\equiv\pm 1\pmod p$
44897
Proof: If $n=ab$ then $2^a-1 \mid 2^n-1$