Suppose that $x \equiv 1 \pmod n$ and $x \equiv 1\pmod m$. Prove that $x \equiv 1 \pmod{ mn}$.
62750
Constant case CRT: $\,x\equiv a\pmod{\! 2},\ x\equiv a\pmod{\! 5}\iff x\equiv a\pmod{\!10}$
149508
Prove $9^n \equiv 9 \pmod{24}\quad \forall n \in \Bbb N^*$
27544
With $a,\ b,\ m,\ n\in \mathbb{Z}$. Show that if ${\rm gcd}(m,n)=1$, then ${\rm gcd}(ma+nb,mn)={\rm gcd}(a,n){\rm gcd}(b,m)$
84733
Why $a\equiv b\pmod n$ implies $a$ and $b$ have equal remainder when divided by $n$?
32762
Proof for: $(a+b)^{p} \equiv a^p + b^p \pmod p$
24820
Prove that if $n$ is a perfect square, then either $n \equiv 1 \pmod 4$, or $n \equiv 0 \pmod 4$.
78356
How to prove ${n \choose p} \equiv\left\lfloor \frac{n}{p} \right\rfloor\pmod p$
94558
Show that $\phi(mn) = \phi(m)\phi(n)\frac{d}{\phi(d)}$
124685
How to show $(mn)!$ is divisible by $(m!)^n$?
21737
If $A|B$ and $B|A$ then prove $A=\pm B$
7151
Let $a,m,n \in \mathbf{N}$. Show that if $\gcd(m,n)=1$, then $\gcd(a,mn)=\gcd(a,m)\cdot\gcd(a,n)$.
151985
If $p$ is a prime and $x,y \in \mathbb{Z}$, then $(x+y)^p \equiv x^p+y^p \pmod{p}$
144017
How to find the smallest $n$ such that $n^a\equiv 1 \pmod p$
100426
Let $a$ and $b$ be integers. Show that if $a \equiv b \pmod{p}$ for every prime $p$, then $a = b$.
111951
Prove $\binom{n}{m}\binom{m}{k} = \binom{n}{k}\binom{n-k}{m-k}$
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
60804
About the property of $m$: if $n < m$ is co-prime to $m$, then $n$ is prime