Two numbers have the divisors 12, 20, 45; their least common multiple is 4320. Determine such pairs of numbers!
Let the pair of sought numbers be denoted by $A, B$. These are multiples of the least common multiple (LCM) of 12, 20, and 45, which is 180. Therefore,

\[
A=180a, \quad B=180b
\]

The LCM of $A$ and $B$ is 4320, which includes 180 as one of its divisors, as well as all common and non-common divisors of $a$ and $b$. This implies that $\frac{4320}{180}=24$ is the LCM of $a$ and $b$. Thus, we need to determine the pairs of numbers $a, b$ whose LCM is 24. Assume that $b \geq a$.

If $b=24$, then $a$ can be any divisor of 24, i.e., any of the numbers

\[
1,2,3,4,6,8,12,24
\]

Thus, if $B=4320$, then $A$ can be any of the numbers

$180,360,540,720,1080,1440,2160,4320$

(8 pairs of numbers!)

If $b=12$, then 24 is the LCM of $a$ and $b$ if $a=2^3=8$ or $a=24$. Since $a \leq b$, only $a=8$ provides a new pair, i.e., $A=1440$ and $B=2160$.

If $b=8$, then 24 is the LCM of $a$ and $b$ if $a=3,6,12,24$. Only $a=3$ and $a=6$ provide new pairs:

\[
A=540, \quad B=1440 \quad \text{and} \quad A=1080 \quad \text{and} \quad B=1440.
\]

Thus, a total of 11 pairs of numbers satisfy the conditions!

Note. We must ensure that $a$ and $b$ are not coprime. The problem does not state that the only divisors of $A$ and $B$ are 12, 20, 45, or their divisors.