For 7.27: Since he can evenly distribute the bills among his nieces, each getting $24$ dollars, the number of bills must be a multiple of $24$, and since he can evenly distribute the bills among his nephews, each getting $40$ dollars, the number of bills must also be a multiple of $40$. Thus, the number of bills must be a multiple of both $24$ and $40$. Any number that is a multiple of both, must be a multiple of their LCM, $120$. Say the number of bills was then $120k$ (for some integer $k$), so how many nieces and how many nephews are there? How many nieces and nephews does he have in total? (The answer to this question will be in terms f $k$). Use this, and the fact that you have $120k$ bills to find out how many bills does each of them get when he distributes them among all of them.

For 7.30: The number $A$ has exactly $9$ divisors. Since we can write $9 = (8+1) = (2 + 1)(2 + 1)$, the number must either be $A = p^8$ for some prime number $p$ (since $9 = 8 + 1$), or of the form $A = p^2 \cdot q^2$ for distinct primes $p$ and $q$ (since $9 = (2 + 1)(2 + 1)$).