Hi Amber! Do you have specific questions about this problems? Is there something you have tried so far but got stuck?

Here are some hints:

7.22: On example 7.2 we could see that for any two numbers $m$ and $n$ it is true that $\gcd(m,n) \times \text{lcm}(m,n) = m\times n$. This problem is exploring if the same happens for any three numbers $l$, $m$, and $n$. Try with different values of $l$, $m$ and $n$ to see if it is true or not.

7.27: Remember a number $n$ is a multiple of both $a$ and $b$ if it is a multiple of $\text{lcm}(a,b)$.

7.28: Look into the prime factorizations of $5$, $510$ and $51$ to find clues of what should be the prime factors in the prime factorizations of $A$, $B$ and $C$. Remember the GCD of two numbers is equal to the product of all the common prime factors of with the smallest exponent you can see in their prime factorizations, and the LCM of two numbers is equal to the product of all distinct prime factors you can see in their prime factorizations with the largest exponent you can see.

7.30: Recall a number $n = p_{1}^{e_1} \cdot p_{2}^{e_2} \cdot \cdots \cdot p_{k}^{e_k}$ has exactly $(e_1 + 1)(e_2 + 1)\cdots(e_k + 1)$ factors. For example, If we want a number to have exactly $25$ factors, since $25 = (24 + 1) = (4 + 1) \times (4 + 1)$, the number should be of the form $p^{24}$ for some prime number $p$, or $p^{4} q^{4}$ for distinct prime numbers $p$ and $q$. What would be the case for numbers that have $9$ and $4$ factors?