Supposing $a^{n}\cong1(mod\,m)$ and $t \cong r(mod\,n)$. How to prove that $a^{t}\cong a^{r}(mod\,m)$?
4972
Mod of numbers with large exponents [modular order reduction]
60777
Prove $\, a_1 \Bbb Z \cap \dotsb \cap a_r \mathbb{Z} = {\rm lcm}(a_1, \ldots, a_r) \Bbb Z$
97469
How to prove $|a^k|=n/\gcd(n,k)$ whenever $|a|=n$?
20161
Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.
141258
Why does $M \mathbin{\otimes_R} N \cong M_\mathfrak{p} \mathbin{\otimes_{R_\mathfrak{p}}} N$?
57654
Congruent Modulo $n$: definition
110703
How Can I prove (a+b) mod m = (a mod m) + (b mod m)) mod m?
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
43990
R is a commutative ring with $1\ne 0$ and $R^m \cong R^n$, then $m=n$
29204
$a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus
145722
Prove there is a bijection between the set of $R[x]$-module structures on $M$ and $\mathrm{End}_{R-\mathsf{Mod}}(M)$.
61825
Congruence class $[a]$ modulo $m$, $\gcd(x, m) = \gcd(a, m)$
155828
How do I show that $A$ and $A^T$ are similar, given that the minimal polynomial of $A$ splits over $\mathbb{F}$?
69967
How to prove $\gcd(a^m-b^m,a^n-b^n) = a^{\gcd(m,n)} - b^{\gcd(m,n)} $?
135986
Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$
81346
Let $r$ be the smallest positive integer such that $a^r \equiv 1 \bmod{n}$. Prove $r \mid \phi(n)$.