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 $p$ be a prime. Show that there exist infinitely many positive integer $n$ such that $p\mid (2^n-n)$.

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

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 $f(m)\ne f(r_1)f(r_2)\cdots f(r_k)$.

Show that there are infinite many composite numbers in the sequence $$1, 31, 331, 3331, 33331, \cdots$$

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

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.

Select nine different digits from $0$ to $9$ to form a two-digit number, a three-digit number and a four-digit number. The sum of these three numbers is $2017$. Which digit is not selected?

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 $10^k\equiv 4\pmod{6}$.

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

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

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 $1^{p}+2^{p}+3^{p}+\cdots+(p-1)^{p}\pmod{p}$.