Starting at the same time, Bill rings a bell every $36$ seconds, and Wendy blows a whistle every $90$ seconds, and Sam blows a horn every $385$ seconds. How many minutes will it be before both Bill and Wendy and Sam simultaneously make a sound?

Answer 1

First, we need to find the least common multiple (LCM) of $36$, $90$, and $385$.

Prime factorizing each number yields:

$36 = 2^2 \cdot 3^2$

$90 = 2 \cdot 3^2 \cdot 5$

$385 = 5 \cdot 7 \cdot 11$

The LCM will be the product of the highest power of each prime factor present in any of the numbers:

LCM $= 2^2 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 = 13860$

Now, to convert $13860$ seconds into minutes:

$13860 \text{ seconds} = 13860 \div 60 \text{ minutes} = 231 \text{ minutes}$

Therefore, it will be $231$ minutes before both Bill, Wendy, and Sam simultaneously make a sound.