a light house flashes every 56 seconds. Another lighthouse flashes every 40 seconds. At 9 p.m. they both flash t the same time. What time will it e when they next both flash at the same time?

first lighthouse will flash in 56 sec, then 112, 168 , etc seconds

second lighthouse will flash in 49 sec, then 80, 120, etc seconds

so this is just like finding the LCD of 56 and 40
56 = 2x2x2x7
40 = 2x2x2x5

LCD = 2x2x2x5x7 = 280
so they will flash together 280 seconds from now or 4 minutes and 40 seconds from 9 pm
which would be 9:04:40 pm

Well, it seems like these lighthouses are playing "catch up" with each other! Let's find out the next time they will flash together.

To determine the next time they will both flash at the same time, we need to find the smallest common multiple of 56 and 40.

The multiples of 56 are: 56, 112, 168, 224, 280, 336, ...

The multiples of 40 are: 40, 80, 120, 160, 200, 240, 280, ...

Ah, there you go! The next time they will both flash simultaneously is at 280 seconds or 4 minutes and 40 seconds.

So, if they start flashing together at 9 p.m., they will next flash at the same time at 9:04:40 p.m.

Now, let's hope they don't get into a flashing competition!

To find the next time when both lighthouses will flash at the same time, we need to determine the least common multiple (LCM) of the two given times: 56 seconds and 40 seconds.

The LCM of 56 and 40 can be found by calculating their prime factors:
56 = 2^3 * 7
40 = 2^3 * 5

To find the LCM, we take the highest powers of all the prime factors involved. Therefore, LCM(56, 40) = 2^3 * 7 * 5 = 280 seconds.

Now, we need to convert 280 seconds into hours and minutes. There are 60 seconds in a minute and 60 minutes in an hour, so:

280 seconds = 280 / 60 minutes = 4 minutes and 40 seconds

Therefore, the next time when both lighthouses will flash at the same time is 9:04 p.m.

To find the time when both lighthouses will next flash at the same time, we need to determine the time it takes for their flashing patterns to align.

Let's calculate the least common multiple (LCM) of 56 seconds and 40 seconds to find the time interval between successive alignments.

The prime factorization of 56 is 2 * 2 * 2 * 7 (or 2^3 * 7), while the prime factorization of 40 is 2 * 2 * 2 * 5 (or 2^3 * 5).

To compute the LCM, we take the highest power of each prime factor occurring in either number, giving us 2^3 * 5 * 7, which is 280.

Therefore, the lighthouses will next both flash at the same time after 280 seconds.

Now, let's convert the seconds to minutes and hours to determine the time.

The time interval of 280 seconds is equal to 280 ÷ 60 = 4 minutes and 40 seconds.

Adding this time to the initial time of 9 p.m., we can find the answer.

9 p.m. + 4 minutes = 9 p.m. + 0 hours 4 minutes = 9:04 p.m.

Hence, the lighthouses will next both flash at the same time at 9:04 p.m.