Find the remainder when $7^{83}$ is divided by $11$.
Find the smallest positive integer $x$ such that $13x \equiv 5 \pmod{17}$.
Compute $\varphi(360)$, where $\varphi$ denotes Euler's totient function.
Find the smallest positive integer $n$ such that $n \equiv 2 \pmod{5}$, $n \equiv 3 \pmod{7}$, and $n \equiv 5 \pmod{8}$.
Convert $2025$ from base $10$ to base $8$. What is the sum of its digits in base $8$?
Find the last three digits of $7^{999}$.
Find the remainder when $3^{100} + 5^{100}$ is divided by $13$.
How many solutions does the congruence $15x \equiv 21 \pmod{36}$ have among $x \in \{0, 1, 2, \ldots, 35\}$?
Find the multiplicative order of $2$ modulo $19$. That is, find the smallest positive integer $k$ such that $2^k \equiv 1 \pmod{19}$.
Find the smallest positive integer $n$ satisfying: $$n \equiv 1 \pmod{3}, \quad n \equiv 2 \pmod{5}, \quad n \equiv 3 \pmod{7}, \quad n \equiv 4 \pmod{11}.$$
Find the largest three-digit number (in base 10) that is a palindrome when written in base 5.
Find the last two digits of $2^{40} + 3^{40} + 7^{40}$.
Find the remainder when $1! + 2! + 3! + \cdots + 50!$ is divided by $21$.
How many integers $n$ in $\{1, 2, \ldots, 100\}$ satisfy $n^2 \equiv 1 \pmod{24}$?
Find the smallest positive integer $N$ satisfying: $$N \equiv 3 \pmod{8}, \quad N \equiv 5 \pmod{9}, \quad N \equiv 7 \pmod{11}.$$