3 bells ring at intervals of 6 12 18 minutes if all the three bells rang at 6 am when will they together again ?

🔔 Problem: Bells Ringing Together

Three bells ring at regular intervals of 6 minutes, 12 minutes, and 18 minutes. If all three bells ring together at 6:00 AM, find when they will ring together again.

📌 Step 1: Understand the Concept

Each bell rings after a fixed interval:

First bell → every 6 minutes
Second bell → every 12 minutes
Third bell → every 18 minutes

To find when they will ring together again, we need to find a common time interval at which all three bells coincide.

👉 This is done using the concept of Least Common Multiple (LCM).

Definition: The LCM of numbers is the smallest number that is divisible by all of them.

📌 Step 2: Find LCM of 6, 12, and 18

Prime Factorization Method:

6 = 2 × 3
12 = 2 × 2 × 3 = 2² × 3
18 = 2 × 3 × 3 = 2 × 3²

Now take the highest power of each prime number:

Highest power of 2 → 2²
Highest power of 3 → 3²

LCM = 2² × 3² = 4 × 9 = 36 minutes

✔ All three bells will ring together after 36 minutes.

📌 Step 3: Find the Exact Time

The bells rang together at 6:00 AM.

Add 36 minutes:

6:00 AM + 36 minutes = 6:36 AM

✅ Final Answer

The three bells will ring together again at 6:36 AM.

💡 Key Learning

When events repeat at fixed intervals, use LCM to find when they coincide.
Always convert intervals into the same unit (here, minutes).
Add the LCM to the starting time to get the final answer.

❓ Why Do We Use LCM and Not HCF in This Problem?

In the given problem, three bells ring at intervals of 6, 12, and 18 minutes, and we are asked when they will ring together again.

📌 Understanding the Situation

Each bell rings repeatedly after a fixed interval:

First bell → every 6 minutes
Second bell → every 12 minutes
Third bell → every 18 minutes

We are looking for a time when all three bells ring at the same moment again.

👉 This means we need a number that is a common multiple of 6, 12, and 18.

✔ Why LCM is Used

The Least Common Multiple (LCM) gives the smallest common time interval after which all events repeat together.

In this case:

Multiples of 6 → 6, 12, 18, 24, 30, 36, …
Multiples of 12 → 12, 24, 36, …
Multiples of 18 → 18, 36, …

👉 The smallest common number in all lists is 36.

✔ So, all bells will ring together again after 36 minutes.

❌ Why NOT HCF?

The Highest Common Factor (HCF) gives the largest number that divides all the numbers.

For 6, 12, and 18:

HCF = 6

But what does 6 represent?

❌ It tells us a common divisor, NOT when events will occur together.

If we used HCF (6 minutes), let’s check:

After 6 minutes → Only first bell rings
Second bell (12 min) ❌ does not ring
Third bell (18 min) ❌ does not ring

👉 So, HCF does not give the correct answer for coincidence of events.

💡 Key Difference

Concept	Meaning	Use Case
LCM	Smallest common multiple	When events repeat together
HCF	Greatest common divisor	When dividing into equal parts

✅ Conclusion

👉 We use LCM because we want the next common time when all bells ring together.

👉 HCF is not suitable because it deals with division, not repetition of events.