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 $(2^n-1)$.

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

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

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 $n^2$ 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 $240$.

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 $24$.

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?

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

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 $N$ be the product of four consecutive odd numbers. Show that $N\equiv 1\pmod{8}$.

Let the product of all odd positive integer not greater than $2019$ be $2019!!$. Find the last three digits of $2019!!$.

Show that there is at least one Friday $13^{th}$ in any year, including any leap year.

Show that there exists an infinite number of integers in the form of $(2^n+27)$ which are multiples of $7$.

In the following $(8\times 5)$ grid, how many shortest routes are there from point $A$ to point $B$?

In the following $5\times 4\times 3$ grid system, how many shortest routes are there from point $A$ to point $B$?

There are $5$ red balls and $4$ green balls in a bag. One ball is retrieved a time until all the balls are taken out. How many possible ways are there such that all the red balls are taken out before all the green balls are taken out?

Let $n$ and $k$ be two positive integers. Show that $$\frac{1}{\binom{n}{k}}=\frac{k}{k-1}\left(\frac{1}{\binom{n-1}{k-1}}-\frac{1}{\binom{n}{k-1}}\right)$$