- we analyze the problem in this way:

- One bell ringing interval is 12 seconds, and the other bell ringing interval is 15 seconds.

- They start and end at the same time ---> their total time is the same, and if the two bell ring together, the result of dividing total time by their interval must be an integer.
  - ex. 72 / 12 = 6, meaning that the first bell has rung 6 times.
     72 / 15 = 4.8, which is impossible ---> the bell must ring an integer times.

- Thus, we need to find the situation where:
  - total time mod 15 = total time mod 12 = 0.

- If the first bell has rung a times, the second bell has rung b times:

12 * a = 15 * b, a : b = 15 : 12 = 5 : 4 - Since we need to get the next time the two bells ring together after they begin, we want the smallest a and b in integer form.

- The answer will be a = 5, b = 4, and the total time is 60s, which is 1 min.

*** Note: I have written a simple code to generalize this problem. If you want to check, please click on this link:
https://github.com/Kexin996/brainteaser/blob/master/july10.py