Question
Brendan's apartment complex and the Monroe Commons apartment complex have the same number of apartments. However, Brendan's apartment complex is built in groups of 11 apartments, while the Monroe Commons apartment complex is built in clusters of 9. What is the smallest number of apartments could each complex have?
Answers
GPT-4o mini
To find the smallest number of apartments that both Brendan's apartment complex and Monroe Commons can have, we need to determine the least common multiple (LCM) of the two numbers: 11 and 9.
First, we note that 11 is a prime number, and it can only be factored into 1 and 11. The number 9 can be factored as \(3^2\) (since \(9 = 3 \times 3\)).
Next, we find the LCM by taking each prime factor raised to the highest power that appears in the factorizations:
- The prime factor 11 appears as \(11^1\).
- The prime factor 3 appears as \(3^2\).
Now we multiply these together to find the LCM:
\[
\text{LCM} = 11^1 \times 3^2 = 11 \times 9
\]
Calculating this:
\[
11 \times 9 = 99
\]
Thus, the least common multiple of 11 and 9 is 99.
Finally, we conclude that the smallest number of apartments each complex could have is
\[
\boxed{99}
\]
First, we note that 11 is a prime number, and it can only be factored into 1 and 11. The number 9 can be factored as \(3^2\) (since \(9 = 3 \times 3\)).
Next, we find the LCM by taking each prime factor raised to the highest power that appears in the factorizations:
- The prime factor 11 appears as \(11^1\).
- The prime factor 3 appears as \(3^2\).
Now we multiply these together to find the LCM:
\[
\text{LCM} = 11^1 \times 3^2 = 11 \times 9
\]
Calculating this:
\[
11 \times 9 = 99
\]
Thus, the least common multiple of 11 and 9 is 99.
Finally, we conclude that the smallest number of apartments each complex could have is
\[
\boxed{99}
\]