Questions: Four tour buses depart the terminal on the same day. The itinerary of the first bus takes 4 days; it takes 10 days for the second bus, 12 days for the third bus, and it takes 20 days for the fourth bus. Assuming that another tour begins the same day that the bus returns to the terminal, how many days will it take until all four buses return to the terminal on the same day?

Solution Steps

To determine when all four buses will return to the terminal on the same day, we need to find the least common multiple (LCM) of the days it takes for each bus to complete its itinerary. The LCM of these numbers will give us the smallest number of days after which all buses will be back at the terminal simultaneously.

The itinerary days for the four tour buses are given as follows:

• Bus 1: \(4\) days
• Bus 2: \(10\) days
• Bus 3: \(12\) days
• Bus 4: \(20\) days

To find when all buses return to the terminal on the same day, we need to calculate the least common multiple (LCM) of the numbers \(4\), \(10\), \(12\), and \(20\).

The LCM can be calculated using the formula: \[ \text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)} \] We apply this iteratively to all four numbers.

After performing the calculations, we find that: \[ \text{LCM}(4, 10, 12, 20) = 60 \]

Final Answer