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 $$(x+y)^p\equiv x^p + y^p\pmod{p}$$

(Fermat's little theorem) Show that $a^p\equiv a\pmod{p}$ holds if $p$ is a prime.

Find all powers of $2$, such that after deleting its first digit, the new number is also a power of 2. For example, $32$ is such a number because $32=2^5$ and $2=2^1$.

An integer in the form of $F_n=2^{2^n}+1$ where integer $n\ge 1$ is called a Fermat's number. Let $d_n$ be any divisor of $F_n$. Show that $d_n\equiv 1\pmod{2^{n+1}}$.

Assume positive integer $n > 1$ satisfies $n\mid (2^n+1)$, prove $n$ is a multiple of $3$.

Let $p$ be an odd prime, and $n=\frac{2^{2p}-1}{3}$ in an integer. Prove $2^{n-1}\equiv 1\pmod{n}$.

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

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

Let integer $N=\left\lfloor{(\sqrt{29}+\sqrt{21})^{2020}}\right\rfloor$ where $\lfloor{x}\rfloor$ denotes the largest integer not exceeding $x$. Find the last two digits of $N$.

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 $\mid 12^m - 5^n\mid$.

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 $2019!!$.

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

Show that there exists an infinite number of squares in the form of $(n\cdot 2^k - 7)$ where $n$ and $k$ are both positive integers.

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 $7$.

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.