GCD and LCM: all problems

Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of $n$?

The least common multiple of $a$ and $b$ is 12, and the least common multiple of $b$ and $c$ is 15. What is the least possible value of the least common multiple of $a$ and $c$?

For how many values of $k$ is $12^{12}$ the least common multiple of the positive integers $6^6$, $8^8$, and $k$?

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 $a$?

(Choices from the original contest: 5, 7, 11, 13, 17.)

If the GCD of 2 numbers is 12 and the LCM of the same 2 numbers is 240, find the product of the numbers.

Two numbers $a$ and $b$ satisfy the condition $\operatorname{lcm}(\gcd(24, a), \gcd(a, b)) = \operatorname{lcm}(\gcd(24, b), \gcd(a, b))$. If $a = 2b$, what is the smallest possible value of $a + b$?

Let $[r, s]$ denote the least common multiple of positive integers $r$ and $s$. Find the number of ordered triples $(a, b, c)$ of positive integers for which $[a, b] = 1000$, $[b, c] = 2000$, and $[c, a] = 2000$.

Three numbers $a$, $b$, and $c$ satisfy the conditions that $\gcd(a, b) = 20$, $\gcd(b, c) = 18$, and $\operatorname{lcm}(a, c) = 5040$. Find the sum of all possible values of $a + c$.

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

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