To find out when both buses will arrive at the station at the same time again, we need to find the least common multiple (LCM) of their arrival intervals.
The two buses come every 15 minutes and 18 minutes.
We can find the LCM of 15 and 18 by using their prime factorization:
- The prime factorization of 15 is \(3 \times 5\).
- The prime factorization of 18 is \(2 \times 3^2\).
To find the LCM, we take the highest power of each prime that appears in the factorizations:
- For \(2\): highest power is \(2^1\) (from 18)
- For \(3\): highest power is \(3^2\) (from 18)
- For \(5\): highest power is \(5^1\) (from 15)
Now, multiply these together:
\[ LCM = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 \]
Calculating this:
\[ 2 \times 9 = 18 \] \[ 18 \times 5 = 90 \]
Thus, the LCM of 15 and 18 is 90.
So, both buses will arrive at the station at the same time again in 90 minutes.