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 sound together in 3 days, we can follow these steps:

### Step 1: Identify the intervals of the alarms
The alarms from the three clocks sound at intervals of:
- Clock 1: Every 2 hours
- Clock 2: Every 4 hours
- Clock 3: Every 6 hours

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

- The prime factorization of each number is:
- 2 = 2^1
- 4 = 2^2
- 6 = 2^1 * 3^1

- The LCM is found by taking the highest power of each prime factor:
- For 2: the highest power is 2^2 (from 4)
- For 3: the highest power is 3^1 (from 6)

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

### Step 3: Calculate the number of times alarms ring together in one day
Since the alarms ring together every 12 hours, we can find out how many times they ring together in a day (24 hours).

- In one day (24 hours), the alarms will ring together:
\[ \frac{24 \text{ hours}}{12 \text{ hours}} = 2 \text{ times} \]

### Step 4: Calculate the number of times alarms ring together in 3 days
Now, to find out how many times they will ring together in 3 days, we multiply the number of times they ring together in one day by 3.

- Total times in 3 days:
\[ 2 \text{ times/day} \times 3 \text{ days} = 6 \text{ times} \]

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

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