Three bells ring at intervals of 4, 6, and 9 minutes, respectively. If they all ring together at 9:00 a.m., at what time will they ring together again?

The correct option is 9:36 a.m.

Step-by-Step Explanation:

Step 1: Understand the Problem
Three bells ring at intervals of 4 minutes, 6 minutes, and 9 minutes, respectively. They ring together at 9:00 a.m. To find out when they will all ring together again, we need to find the smallest common multiple of their time intervals, which is the Least Common Multiple (LCM) of 4, 6, and 9.

Step 2: Find the Prime Factorization
Express each interval as the product of its prime factors:

4 = 2²

6 = 2 × 3

9 = 3²

Step 3: Calculate the LCM
The LCM is obtained by taking the highest power of each prime factor involved:

The highest power of 2 is 2² = 4.

The highest power of 3 is 3² = 9.

LCM(4, 6, 9) = 2² × 3² = 4 × 9 = 36 minutes.

Step 4: Determine the Next Simultaneous Time
The bells will ring together again after 36 minutes from their initial simultaneous ring at 9:00 a.m.

9:00 a.m. + 36 minutes = 9:36 a.m.

Therefore, the bells will ring together again at 9:36 a.m.