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Let $R$ be the set of all possible remainders when a number of the form $2^n$, where $n$ is a non-negative integer, is divided by $1000$. Let $S$ be the sum of the elements in $R$. Find the remainder when $S$ is divided by $1000$.

What is the smallest positive integer $n$ such that $20\equiv n^{15} \pmod{29}$?

Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3\le p<100$, and $1\le a < p$ such that the sum $a+a^2+a^3+ \cdots + a^{(p-2)!}$ is not divisible by $p$?

Let sequence $g(n)$ satisfy $g(1)=0, g(2)=1, g(n+2)=g(n+1)+g(n)+1$ where $n\ge 1$. Show that if $n$ is a prime greater than 5, then $n\mid g(n)[g(n)+1]$.

Find the least non-negative residue of $70! \pmod{5183}$.

Compute $50^{250} \pmod{83}$ .

Show that if $n$ is an integer greater than $1$, then $(2^n-1)$ is not divisible by $n$.

Let $x$ be an integer and $p$ is a prime divisor of $(x^6 + x^5 + \cdots + 1)$. Show that $p=7$ or $p\equiv 1\pmod{7}$.

Let $p$ be an odd prime divisor of number $(a^2+1)$ where $a$ is an integer. Show that $p\equiv 1\pmod{4}$.

Compute $3^{2018} \mod{17}$.

Compute $\underbrace{3^{3^{3^{\cdots^{3}}}}}_{2012\ times}\pmod{100}$.

Solve $15x\equiv 7\pmod{32}$.

Solve $x^{12}\equiv 3\pmod{11}$.

Show that $x^5\equiv 3\pmod{11}$ is not solvable.

Find one solution to $x^7\equiv 3\pmod{11}$.

Show that $(2^{1194} + 1)$ is a multiple of $65$.

Solve $x^{22} + x^{11}\equiv 2\pmod{11}$.

Compute $20!\pmod{23}$.

Let $p$ be a prime and integer $a$ is co-prime to $p$, show that $$a^{p(p-1)}\equiv 1\pmod{p^2}$$

Let $p$ and $q$ be two distinct primes, and integer $a$ is co-prime to both $p$ and $q$, show $$a^{(p-1)(q-1)}\equiv 1\pmod{pq}$$

Show that for any positive integer $n$, $\varphi(2^n-1)$ is a multiple of $n$ where $\varphi(n)$ is Euler's totient function.

Let $p$ be an odd prime divisor of integer $(n^4 + 1)$. Show that $p\equiv 1\pmod{8}$.

Show that a prime $p > 2$ is a sum of two squares if and only if $p\equiv 1\pmod{4}$.

(Two Squares Theorem) Show that a positive integer $n$ is a sum of two squares if and only if each prime factor $p$ of $n$ such that $p\equiv 3\pmod{4}$ occurs to an even power in the prime factorization of $n$.

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