Three bells ring at intervals of 15,24 and 36 minutes. The bell rings together at 2:00 p.m. . At what time they will ring together again​

Answer :

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Answer:

To determine when the three bells will ring together again, we need to find the least common multiple (LCM) of their ringing intervals: 15 minutes, 24 minutes, and 36 minutes.

**Step 1: Prime Factorization**

First, we find the prime factors of each interval:

- \( 15 = 3 \times 5 \)

- \( 24 = 2^3 \times 3 \)

- \( 36 = 2^2 \times 3^2 \)

**Step 2: Identify the Highest Powers of Each Prime**

Next, we identify the highest powers of each prime number that appear in these factorizations:

- The highest power of \( 2 \) is \( 2^3 \) (from 24).

- The highest power of \( 3 \) is \( 3^2 \) (from 36).

- The highest power of \( 5 \) is \( 5 \) (from 15).

**Step 3: Calculate the LCM**

The LCM is found by multiplying these highest powers together:

{LCM} = 2^3 \times 3^2 \times 5

Calculating this:

2^3 = 8

3^2 = 9

8 \times 9 = 72

72 \times 5 = 360

So, the LCM of 15, 24, and 36 is 360 minutes.

**Step 4: Determine the Time**

Since the bells ring together at 2:00 PM, we add 360 minutes to this time to find when they will ring together again.

360 \text{ minutes} = 6 \text{ hours}

Adding 6 hours to 2:00 PM:

2:00 \, \text{PM} + 6 \, \text{hours} = 8:00 \, \text{PM}

Therefore, the three bells will ring together again at 8:00 PM.

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