Intervals 3487

The four teams left the terminal together at 5:00 in the morning. Line A runs at 15-minute intervals, line B at 6-minute intervals, line C at 20-minute intervals, and line D at 8-minute intervals.
What time did all four lines leave the terminal together again?
How many times in the morning, t. j. from 5:00 to 12:00, all lines are based on the final together?

Result

a = 7:00 hh:mm
n =  3

Step-by-step explanation:

$$\begin{gathered} 15=3\cdot 5 \\ 6=2\cdot 3 \\ 20=2^2\cdot 5 \\ 8=2^3 \\ LCM(15,6,20,8)=2^3\cdot 3\cdot 5=120 \\ d=LCM(15,6,20,8)=120 \\ a=5+d/60=5+120/60=7=7:00\text{hh}:\text{mm} \end{gathered}$$

$$\begin{gathered} m=(12.00-5.00)/(d/60)=(12.00-5.00)/(120/60)=\frac{7}{2}=3\frac{1}{2}=3.5 \\ n=\lfloor m\rfloor =\lfloor 3.5\rfloor =3 \end{gathered}$$

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