Four clocks sound at the intervals of 3,5,12& 15 min s. All the four sounded together at 12 o'clock noon,now at what time will they sound together again???
What is the least common multiple of the four intervals?
To find the time when four clocks will sound together again, we need to find the least common multiple (LCM) of the intervals at which they sound.
The intervals at which the clocks sound are 3, 5, 12, and 15 minutes. To find the LCM, we can follow these steps:
Step 1: List the prime factors of each interval:
- 3 = 3 (prime number)
- 5 = 5 (prime number)
- 12 = 2^2 * 3
- 15 = 3 * 5
Step 2: Identify the highest power of each prime factor across all the intervals:
- The highest power of 2 is 2^2 = 4.
- The highest power of 3 is 3.
- The highest power of 5 is 5.
Step 3: Multiply the highest powers of all the prime factors:
- 2^2 * 3 * 5 = 4 * 3 * 5 = 60
Therefore, the LCM of the intervals is 60 minutes.
Since the clocks sounded together at 12 o'clock noon, the next time they will sound together is after 60 minutes, which is 1 hour. So, they will sound together again at 1 o'clock in the afternoon.