4. [HCF/LCM reasoning] Two bells ring at intervals of $18$ minutes and $24$ minutes respectively. If they ring together at $9:00$ a.m., at what time will they next ring together?

Step 1: Understand the Problem

To find when the bells will ring together again, we need to find the smallest common multiple of their individual ringing intervals. This is precisely what the Least Common Multiple (LCM) helps us determine.

Step 2: Prime Factorization of 18

First, we find the prime factorization of 18. This means expressing 18 as a product of its prime factors. $18 = 2 \times 9 = 2 \times 3 \times 3 = 2 \times 3^2$.

Step 3: Prime Factorization of 24

Next, we find the prime factorization of 24. $24 = 2 \times 12 = 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3$.

Step 4: Calculate the LCM

To find the LCM, we take the highest power of each prime factor present in the factorizations of 18 and 24. The prime factors are 2 and 3. The highest power of 2 is $2^3$ (from 24) and the highest power of 3 is $3^2$ (from 18). Multiplying these together gives us the LCM.

Step 5: Determine the Next Ringing Time

The LCM, 72 minutes, is the time after which the bells will ring together again. We convert 72 minutes into hours and minutes ($72 \text{ minutes} = 1 \text{ hour and } 12 \text{ minutes}$) and add this duration to the initial ringing time of 9:00 a.m.

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