Alarms from 3 different clocks sound after every 2, 4 and 6 hours, respectively. If the clocks are started at the same time, how many times do the alarms ring together in 3 days?

Options

A. 6
B. 3
C. 9
D. 2

Answer

To solve the problem of how many times the alarms from three different clocks ring together in 3 days, we can follow these steps:

### Step 1: Determine the ringing intervals of the clocks
The three clocks ring at intervals of:
- Clock 1: every 2 hours
- Clock 2: every 4 hours
- Clock 3: every 6 hours

### Step 2: Calculate the Least Common Multiple (LCM) of the intervals
To find out when all three alarms will ring together, we need to calculate the LCM of the intervals (2, 4, and 6).

- Prime factorization:
- 2 = 2^1
- 4 = 2^2
- 6 = 2^1 × 3^1

- To find the LCM, take the highest power of each prime factor:
- For 2: highest power is 2^2 (from 4)
- For 3: highest power is 3^1 (from 6)

Thus, the LCM = 2^2 × 3^1 = 4 × 3 = 12 hours.

### Step 3: Determine how many times the alarms ring together in 3 days
Now that we know the alarms ring together every 12 hours, we need to find out how many times this occurs in 3 days.

- Calculate the total hours in 3 days:
- 1 day = 24 hours
- 3 days = 3 × 24 = 72 hours

### Step 4: Calculate the number of times the alarms ring together
To find out how many times the alarms ring together in 72 hours, we divide the total hours by the interval at which they ring together:

- Number of times = Total hours / LCM
- Number of times = 72 hours / 12 hours = 6 times.

### Final Answer
The alarms ring together **6 times** in 3 days.

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