The traffic lights at three different road crossings change every 48 seconds, 72 seconds, and 108 seconds, respectively. To find out when they will change simultaneously again, we need to find the least common multiple (LCM) of these three time intervals.
First, let’s break down each number into its prime factors:
- 48 = 24 × 31
- 72 = 23 × 32
- 108 = 22 × 33
To find the LCM, we take the highest power of each prime number that appears in the factorization:
- For the prime number 2, the highest power is 24 (from 48).
- For the prime number 3, the highest power is 33 (from 108).
Now, we calculate the LCM:
LCM = 24 × 33 = 16 × 27 = 432.
This means that the three traffic lights will change simultaneously again after 432 seconds.
To convert this into minutes, we divide by 60:
432 seconds ÷ 60 = 7 minutes and 12 seconds.
Therefore, all three traffic lights will change simultaneously again after 7 minutes and 12 seconds.