Starting at the same time, Bill rings a bell every $36$ seconds, and Wendy blows a whistle every $1000$ seconds. How many minutes will it be before both Bill and Wendy simultaneously make a sound?

Answer 1

To find the least common multiple (LCM) of $36$ and $1000$, we can first find the prime factorization of each number:

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

$1000 = 2^3 \cdot 5^3$

The LCM is the highest power of each prime that appears in the prime factorization of both numbers. Therefore, the LCM of $36$ and $1000$ is $2^3 \cdot 3^2 \cdot 5^3 = 8 \cdot 9 \cdot 125 = 9000$.

Since there are $60$ seconds in a minute, it will take $\frac{9000}{60} = 150$ minutes before both Bill and Wendy simultaneously make a sound.