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 $3x\equiv 5\pmod{8}$.

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

Let sequence $\{x_n\}$ satisfy the relation $x_{n+2}=x_{n+1}+2x_n$ for $n\ge 1$ where $x_1=1$ and $x_2=3$.

Let sequence $\{y_n\}$ satisfy the relation $y_{n+2}=2y_{n+1}+3y_n$ for $n\ge 1$ where $y_1=7$ and $y_2=17$.

Show that these two sequences do not share any common term.

Compute $9^{50}\pmod{1000}$.

Find the last three digits of $9 + 9^2 + 9^3 + \cdots + 9^{2000}$.

Let $N = 7\times 8\times 9\times 15\times 16\times 17\times 23\times 24\times 25\times 43$. Compute $N\pmod{11}$.

Show that $\varphi(n)=n/4$ is impossible to hold.

Show that from any given $m$ integers, it is always possible to select one or more integers such that their sum is a multiple of $m$.

Find the remainder when $10^{10}+10^{100}+10^{1000}+\cdots+10^{\overbrace{\scriptsize{10\cdots 0}}^{2018}}$ is divided by $7$.

How many positive integers $N$, less than $2017$, satisfy $$N^{2016^{2016}}\equiv 1\pmod{2017}$$

Let $p$ is an odd prime, compute $1^{p-1}+2^{p-1}+3^{p-1}+\cdots+(p-1)^{p-1}\pmod{p}$.

Let $p$ is an odd prime, compute $1^{p}+2^{p}+3^{p}+\cdots+(p-1)^{p}\pmod{p}$.

Let $m$ and $n$ be two distinct positive integers. Find the minimal value of $(m+n)$ such that the last three digits of $2017^m$ and $2017^n$ are equal.

Find the multiplicative order of $17$ modulo $1000$.

Let $p$ be a prime and $k$ be a positive integer less than $p$. Show that $\binom{p}{k} \equiv 0 \pmod{p}$.

Let $x$ and $y$ be two integers and $p$ be a prime. Show that $$(x+y)^p\equiv x^p + y^p\pmod{p}$$

Solve this modular equation: $$f(x)=4x^2+27x-9\equiv 0\pmod{15}$$

Compute $3^{2017}\pmod{1000}$.

Let $n$ be a positive integer and $k$ be an odd positive integer, show $k^{2^n}\equiv 1\pmod{2^{n+2}}$.

Let $S$ be the sum of squares of $10$ consecutive positive integers. Show $S$ cannot be a square.

Let $\{ a_1, a_2, \cdots, a_{2n+1}\}$ be a set of integers such that after removing any element, the remaining ones can always be equally divided into two groups with equal sum. Show that all these $a_i$, $(1 \le i \le 2n+1)$ are equal.

Let $N=4568^{7777}$, $a$ be the sum of digits in $N$, $b$ be the sum of digits in $a$, and $c$ be the sum of digits in $b$. Find $c$.