7. Lighthouse Alpha and Lighthouse Bravo flash their lights every 24 seconds and 30 seconds respectively. Given that they flash together at \( 9.15 \mathrm{p} . \mathrm{m} \)., how long will it take the lighthouses to flash their lights together again?

To find the next time the lighthouses will flash together, we need to determine the least common multiple (LCM) of their flashing intervals: 24 seconds and 30 seconds. The prime factorization of 24 is \(2^3 \times 3^1\) and for 30 it is \(2^1 \times 3^1 \times 5^1\). The LCM is found by taking the highest powers of each prime factor involved: - For 2, the highest power is \(2^3\) - For 3, the highest power is \(3^1\) - For 5, the highest power is \(5^1\) Calculating the LCM: \[ LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 = 120 \text{ seconds} \] This means the lighthouses will flash together again after 120 seconds (or 2 minutes). If they flash together at \(9:15 \text{ p.m.}\), adding 2 minutes gives us \(9:17 \text{ p.m.}\). So, the lighthouses will flash together again at \(9:17 \text{ p.m.}\).