Digital Watch Beeping Synchronization Explained

This problem involves finding the next time two events, occurring at regular intervals, will happen simultaneously. This is a classic application of the Least Common Multiple (LCM).

Watch Beeping Intervals

We are given the beeping intervals for two digital watches, Watch X and Watch Y:

• Watch X beeps every 30 seconds.
• Watch Y beeps every 32 seconds.

They both beeped together at 10:00 AM. To find the next time they will beep together, we need to find the smallest common multiple of their beeping intervals.

Calculating the Least Common Multiple (LCM)

The Least Common Multiple (LCM) of two numbers is the smallest positive integer that is divisible by both numbers. We will find the LCM of 30 and 32 to determine how many seconds will pass before they beep together again.

Prime Factorization of Intervals

First, let's find the prime factorization of each beeping interval:

• For 30 seconds:
  • \(30 = 2 \times 15\)
  • \(30 = 2 \times 3 \times 5\)
• For 32 seconds:
  • \(32 = 2 \times 16\)
  • \(32 = 2 \times 2 \times 8\)
  • \(32 = 2 \times 2 \times 2 \times 4\)
  • \(32 = 2 \times 2 \times 2 \times 2 \times 2\)
  • \(32 = 2^5\)

Finding the LCM

To find the LCM, we take the highest power of all prime factors present in either number:

• Prime factors are 2, 3, and 5.
• Highest power of 2 is \(2^5\) (from 32).
• Highest power of 3 is \(3^1\) (from 30).
• Highest power of 5 is \(5^1\) (from 30).

So, the LCM is:

\(\text{LCM}(30, 32) = 2^5 \times 3^1 \times 5^1\)

\(\text{LCM}(30, 32) = 32 \times 3 \times 5\)

\(\text{LCM}(30, 32) = 32 \times 15\)

\(\text{LCM}(30, 32) = 480\) seconds

Converting Seconds to Minutes

Since time is often expressed in minutes and hours, let's convert 480 seconds into minutes:

\(\text{Minutes} = \frac{\text{Total seconds}}{\text{Seconds per minute}}\)

\(\text{Minutes} = \frac{480}{60}\)

\(\text{Minutes} = 8\) minutes

This means that Watch X and Watch Y will beep together every 8 minutes.

Determining the Next Beep Time

The watches beeped together initially at 10:00 AM. To find the immediate next time they will beep together, we add the calculated LCM duration (8 minutes) to the initial time:

\(\text{Next Beep Time} = \text{Initial Beep Time} + \text{LCM in minutes}\)

\(\text{Next Beep Time} = 10:00 \text{ AM} + 8 \text{ minutes}\)

\(\text{Next Beep Time} = 10:08 \text{ AM}\)

Therefore, the immediate next time that the digital watches X and Y will beep together is 10:08 AM.