Four bells ring 12,18,26,30 times in a sec . After how many minutes they will ring togather

Answer

To solve the problem of when the four bells will ring together, we need to find the least common multiple (LCM) of the number of times each bell rings in a second: 12, 18, 26, and 30. Here’s a step-by-step solution:

### Step 1: Find the prime factorization of each number.
- **12**: \( 12 = 2^2 \times 3^1 \)
- **18**: \( 18 = 2^1 \times 3^2 \)
- **26**: \( 26 = 2^1 \times 13^1 \)
- **30**: \( 30 = 2^1 \times 3^1 \times 5^1 \)

### Step 2: Identify the highest power of each prime factor.
- For \(2\): The highest power is \(2^2\) (from 12).
- For \(3\): The highest power is \(3^2\) (from 18).
- For \(5\): The highest power is \(5^1\) (from 30).
- For \(13\): The highest power is \(13^1\) (from 26).

### Step 3: Multiply the highest powers together to find the LCM.
\[ \text{LCM} = 2^2 \times 3^2 \times 5^1 \times 13^1 \]
Calculating this step-by-step:
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(5^1 = 5\)
- \(13^1 = 13\)

Now, multiply these values:
\[ \text{LCM} = 4 \times 9 = 36 \]
\[ 36 \times 5 = 180 \]
\[ 180 \times 13 = 2340 \]

### Step 4: Convert the LCM from seconds to minutes.
Since the LCM is in seconds, we need to convert it to minutes:
\[ \text{Minutes} = \frac{2340 \text{ seconds}}{60} = 39 \text{ minutes} \]

### Final Answer:
The four bells will ring together after **39 minutes**.

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