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?
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Step-by-Step Solution
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=2×9=2×3×3=2×32.
Step 3: Prime Factorization of 24
Next, we find the prime factorization of 24. 24=2×12=2×2×6=2×2×2×3=23×3.
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 23 (from 24) and the highest power of 3 is 32 (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 minutes=1 hour and 12 minutes) and add this duration to the initial ringing time of 9:00 a.m.