Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were $71$, $76$, $80$, $82$, and $91$. What was the last score Mrs. Walter entered.

Let $\mathbb{S}$ be the set of integers between $1$ and $2^{40}$ that contain two $1$s when written in base $2$. What is the probability that a random integer from $\mathbb{S}$ is divisible by $9$?

Find the multiplicative order of $3$ modulo $17$.

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 $(2^n-1)$.

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 $p\equiv 1\pmod{8}$.

(Thue's theorem) Let $p$ be a prime. Show that for any integer $a$ such that $p\not\mid a$, there exist positive integers $x$, $y$ not exceeding $\lfloor{\sqrt{p}}\rfloor$ satisfying $ax\equiv y\pmod{p}$ or $ax\equiv -y\pmod{p}$.

Show that if the equation $a^2 + 1\equiv 0\pmod{p}$ is solvable for some $a$, then $p$ can be represented as a sum of two squares.

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 $125$.

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

Find the smallest integer $N$ such that $\varphi(n) \ge 5$ holds for all integer $n \ge N$.

Show that two positive integers $m$ and $n$ are co-prime if and only if $\varphi(mn)=\varphi(m)\varphi(n)$.

Solve the system of congruence $$\left\{ \begin{array}{l} x\equiv 1\pmod{3}\\ x\equiv 2\pmod{5}\\ x\equiv 3\pmod{7} \end{array}  \right.$$

Find the multiplicative order of $3$ modulo $301$.

Solve the congruent system: $4x\equiv 2\pmod{6}$ and $3x\equiv 5\pmod{8}$.

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 $27 \mid (x+y+z)$