GCD & LCM: all problems

Let $a, b, c$, and $d$ be positive integers such that $\gcd(a, b) = 24$, $\gcd(b, c) = 36$, $\gcd(c, d) = 54$, and $70 < \gcd(d, a) < 100$. Which of the following must be a divisor of $a$?

Prove that the fraction $\dfrac{21n+4}{14n+3}$ is irreducible for every natural number $n$.

What is the largest positive integer $n$ for which $n^{3} + 100$ is divisible by $n + 10$?

How many ordered pairs $(a, b)$ of positive integers satisfy the equation

where $\gcd(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $\operatorname{lcm}(a, b)$ denotes their least common multiple?

Suppose $a$, $b$, and $c$ are positive integers such that

Which of the following statements are necessarily true?

I. If $\gcd(a, 14) = 1$ or $\gcd(b, 15) = 1$ or both, then $\gcd(c, 210) = 1$.

II. If $\gcd(c, 210) = 1$, then $\gcd(a, 14) = 1$ or $\gcd(b, 15) = 1$ or both.

III. $\gcd(c, 210) = 1$ if and only if $\gcd(a, 14) = \gcd(b, 15) = 1$.

Suppose that $a, b, c$, and $d$ are positive integers satisfying all of the following relations:

What is $\gcd(a, b, c, d)$?

Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$?

There are $10$ horses, named Horse $1$, Horse $2$, $\ldots$, Horse $10$. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T > 0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$?

Suppose $a$, $b$, $c$ are positive integers such that

and

What is the sum of all possible distinct values of $a^{2} + b^{2} + c^{2}$?

Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$?

Let $n$ be the smallest positive integer such that $n$ is divisible by $20$, $n^{2}$ is a perfect cube, and $n^{3}$ is a perfect square. What is the number of digits of $n$?

Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1, 2, \ldots, \dfrac{n}{2}, n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?

How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n$?

For how many ordered pairs of positive integers $(x, y)$, with $y < x \le 100$, are both $\dfrac{x}{y}$ and $\dfrac{x+1}{y+1}$ integers?

Let $n$ be the least positive integer greater than $1000$ for which

What is the sum of the digits of $n$?

Consider the sequence $(a_k)_{k \ge 1}$ of positive rational numbers defined by $a_1 = \dfrac{2020}{2021}$ and for $k \ge 1$, if $a_k = \dfrac{m}{n}$ for relatively prime positive integers $m$ and $n$, then

Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\dfrac{t}{t+1}$ for some positive integer $t$.

Find the number of ordered pairs $(m, n)$ such that $m$ and $n$ are positive integers in the set $\{1, 2, \ldots, 30\}$ and the greatest common divisor of $2^{m} + 1$ and $2^{n} - 1$ is not $1$.