Answers

1. Least common multiple

   The least common multiple of two or more natural numbers is the least common multiple of all of them. This concept has historically been linked to natural numbers, but can be used for negative integers or complex numbers.

   We calculate the least common multiple, by simultaneous decomposition, this method consists of extracting the common and non-common prime factors, therefore

   The lcm of 12,15, 18

   • 12  –  15  –  18 | 2
   • 6  –  15  –   9 | 2
   • 3  –  15  –   9 | 3
   •  1  –   5  –   3  | 3
   •  1  –   5  –   1  | 5
   •  1  –   1  –    1  |

   L.c.m.(12,15,18)= 2² × 3² × 5 = 180 min

   The least common multiple of 12, 15, and 18 is 180.

   Convert the minutes to hours, for this we apply the rule of 3:

   x = 180 * 1 / 60 = 3 hr

   As the bells all together ring at 6 am, so we add

   6 a.m + 3 = 9

   Answer: The bells are rung together again at 9 in the morning.

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