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 $$a^{(p-1)(q-1)}\equiv 1\pmod{pq}$$

Let $p$ be an odd prime. Show that $$\sum_{j=0}^p\binom{p}{j}\binom{p+j}{j}\equiv 2^p +1 \pmod{p^2}$$

Given $30!$ ends with some zeros, what is the digit that immediately precedes these zeros?

The two-digit integers from $19$ to $92$ are written consecutively to form the large integer $$N=192021\cdots 909192$$

Suppose that the $3^k$ is the highest power of $3$ that is a factor of $N$. What is $k$.

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

Show that if there exist integer $x$, $y$, and $z$ such that $3^x + 4^y=5^z$, then both $x$ and $z$ must be even.