Step 1: Conceptual Basis:
To determine when traffic lights will synchronize, the least common multiple (LCM) of their individual cycle times is required. The LCM represents the shortest duration until all cycles align again.
Step 2: Methodology:
Compute the LCM for the intervals of 48, 72, and 108.
Step 3: Calculation and Derivation:
First, obtain the prime factorization for each number:
\[ 48 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3^1 \]\[ 72 = 2 \times 36 = 2 \times 2 \times 18 = 2 \times 2 \times 2 \times 9 = 2^3 \times 3^2 \]\[ 108 = 2 \times 54 = 2 \times 2 \times 27 = 2^2 \times 3^3 \]The LCM is calculated by multiplying the highest power of each unique prime factor found across the numbers.
\[ \text{LCM}(48, 72, 108) = 2^4 \times 3^3 \]\[ \text{LCM} = 16 \times 27 = 432 \text{ seconds} \]This indicates that the lights will coincide every 432 seconds.
Next, convert 432 seconds to minutes and seconds:
\[ \frac{432}{60} = 7 \text{ with a remainder of } 12 \]Therefore, 432 seconds is equivalent to 7 minutes and 12 seconds.
The initial synchronized change occurred at 8:20:00 hours.
The subsequent synchronized change will take place after an interval of 7 minutes and 12 seconds.
New time = 8 hours : 20 minutes : 00 seconds + 7 minutes : 12 seconds
New time = 8:27:12 hours.
Step 4: Conclusion:
The traffic lights will next synchronize at 8:27:12 hours.