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 $n$. $$2-1, 2^2 -1, \cdots, 2^{n-1} -1$$

Let $n^2$ be a square. Show that $n^2\equiv 0, 1\pmod{3}$.

Let $a$, $b$, $c$, and $d$ be four positive integers. Show that $\left(a^{4b+d}-a^{4c+d}\right)$ must be a multiple of $240$.

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.

Let $n$ be an odd integer greater than $3$, and $\mathbb{S}=\{0, 1, \cdots, n-1\}$. Show that after removing any element from $\mathbb{S}$, it is always possible to equally divide the remaining elements in $\mathbb{S}$ into two groups such that their sum are congruent modulo $n$.

Let $n$ be a positive integer not less than $4$. Show that there exists a polynomial with integral coefficients $$f(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2}+\cdots + a_1 x + a_0$$

such that for any positive integer $m$ and any $k \ge 2$ distinct integers $r_1$, $r_2$, $\cdots$, $r_k$, it always hold that $f(m)\ne f(r_1)f(r_2)\cdots f(r_k)$.

Solve this equation in integers: $x_1^4 + x_2^4 + \cdots + x_{14}^4 = 9999$.

Suppose integers $a$ and $b$ satisfy $ab\equiv -1 \pmod{24}$. Prove $(a + b)$ must be a multiple of $24$.

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

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

Let $p$ be a prime and $$\frac{a}{b}=\frac{1}{1^2}+\frac{1}{2^2}+\cdots + \frac{1}{(p-1)^2}$$

where $a$ and $b$ are two co-prime positive integers. Show that $p\mid a$.

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

Let $p$ be a prime number and $\lfloor{x}\rfloor$ denote the largest integer not exceeding real number $x$. Show that $$C_n^p\equiv\left\lfloor{\frac{n}{p}}\right\rfloor\pmod{p}$$

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 $m$.

Show that for any positive integer $k$, it always holds that $10^k\equiv 4\pmod{6}$.

Solve the following relation in integers: $$x^2 + a^2 = (x+1)^2 + b^2 = (x+2)^2 + c^2 = (x+3)^2 + d^2$$

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

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

Find the largest integer $x$ such that for any positive integer $y$, the number $(7^y + 12y-1)$ is always a multiple of $x$.

Let $m$ and $n$ be two positive integers, find the minimal value of $\mid 12^m - 5^n\mid$.