Find the multiplicative order of $3$ modulo $17$.

Find the multiplicative order of $5$ modulo $19$.

Show that if integer $a$ has multiplicative order of $hk$ modulo $n$, then $a^h$ has order of $k$ modulo $n$.

Let $p$ be an odd prime, and integer $a$ has multiplicative order of $2k$ modulo $p$, then $a^k\equiv -1\pmod{p}$.

Let $n$ be an odd integer greater than $1$, then $n$ is the multiplicative order of $2$ modulo $(2^n-1)$.

Show that for any positive integer $n$, $\varphi(2^n-1)$ is a multiple of $n$ where $\varphi(n)$ is Euler's totient function.

Let $p$ be an odd prime divisor of integer $(n^4 + 1)$. Show that $p\equiv 1\pmod{8}$.

Show that if there exist integer $x$, $y$, and $z$ such that $3^x + 4^y=5^z$, then both $x$ and $z$ must be even.

(Thue's theorem) Let $p$ be a prime. Show that for any integer $a$ such that $p\not\mid a$, there exist positive integers $x$, $y$ not exceeding $\lfloor{\sqrt{p}}\rfloor$ satisfying $ax\equiv y\pmod{p}$ or $ax\equiv -y\pmod{p}$.

Show that if the equation $a^2 + 1\equiv 0\pmod{p}$ is solvable for some $a$, then $p$ can be represented as a sum of two squares.

Show that a prime $p > 2$ is a sum of two squares if and only if $p\equiv 1\pmod{4}$.

Let $a$ and $b$ be two positive integers such that both of them can be written as a sum of two squares. Show that their product can be written as a sum of two squares in two ways.

(Two Squares Theorem) Show that a positive integer $n$ is a sum of two squares if and only if each prime factor $p$ of $n$ such that $p\equiv 3\pmod{4}$ occurs to an even power in the prime factorization of $n$.

Find the multiplicative order of $2$ modulo $125$.

Calculate $3^{64}\pmod{67}$.

Find the smallest integer $N$ such that $\varphi(n) \ge 5$ holds for all integer $n \ge N$.

Show that two positive integers $m$ and $n$ are co-prime if and only if $\varphi(mn)=\varphi(m)\varphi(n)$.

Solve the system of congruence $$\left\{ \begin{array}{l} x\equiv 1\pmod{3}\\ x\equiv 2\pmod{5}\\ x\equiv 3\pmod{7} \end{array}  \right.$$

Find the multiplicative order of $3$ modulo $301$.

Solve the congruent system: $4x\equiv 2\pmod{6}$ and $3x\equiv 5\pmod{8}$.

Find the smallest positive integer $n$ such that $$\left\{  \begin{array}{l} n\equiv 1\pmod{3} \\ n\equiv 3\pmod{5} \\ n\equiv 5\pmod{7} \end{array} \right.$$

Let $n$ be an integer greater than $1$. If none of $1!$, $2!$, $\cdots$, $n!$ has the same remainder when being divided by $n$, show that $n$ is a prime.

Let integers $x$, $y$, $z$ satisfy $$(x-y)(y-z)(z-x)=x+y+z$$

Show that $27 \mid (x+y+z)$

Let $n$ be a positive odd integer. Show that at least one of the following numbers is a multiple of $n$. $$2-1, 2^2 -1, \cdots, 2^{n-1} -1$$