How many positive divisors does $360$ have?
Find the sum of all prime numbers $p$ satisfying $20 \le p \le 50$.
A positive integer $n$ has the form $n = 2^a \cdot 3^b$ (where $a, b$ are positive integers), has exactly $12$ positive divisors, and is divisible by $18$. Find the smallest such $n$.
Find the largest prime factor of $2^{12} - 1$.
Positive integers $a$ and $b$ satisfy $\gcd(a, b) = 15$ and $\text{lcm}(a, b) = 630$. How many ordered pairs $(a, b)$ with $a \le b$ are there?
Find $\gcd(3^{15} - 1,\; 3^{10} - 1)$.
Find the remainder when $3^{200}$ is divided by $13$.
Find the last two digits of $7^{50}$ (i.e., find the remainder when $7^{50}$ is divided by $100$).
Find the sum of all positive divisors of $120$.
Find the remainder when $2^{2024}$ is divided by $37$.
How many positive integers $n \le 100$ satisfy the property that both $n$ and $n + 2$ are prime?
Find the number of positive integers $n \le 100$ such that $\gcd(n, 100) = 5$.
Find the last three digits of $13^{100}$ (i.e., compute $13^{100} \pmod{1000}$).
Let $S$ be the set of all positive divisors of $N = 2^4 \cdot 3^3 \cdot 5^2$. How many ordered pairs $(a, b)$ with $a, b \in S$ satisfy $\text{lcm}(a, b) = N$?
Find the smallest positive integer $n$ such that $n!$ is divisible by $2^{10} \cdot 3^5 \cdot 5^2 \cdot 7$.