The next time they will flash again will be #n# seconds after #9:00:00# pm, where #n# is the least common multiple (LCM) of #56# and #40#. To find the LCM, we look at the prime factorizations of the two values and take the least product of primes which is divisible by both.
#56 = 2^3xx7#
#40 = 2^3 xx 5#
We can see that the LCM requires at least three factors of #2#, one factor of #5#, and one factor of #7#. Thus
#LCM(40, 56) = 2^3xx5xx7 = 280#.
As #280# seconds is equal to #4# minutes and #40# seconds, we can add that value to #9:00:00# pm to find that the next time they flash at the same time is at #9:04:40#pm.
Note that at that time, the first lighthouse will have flashed #280/56 = 5# times, and the second will have flashed #280/40 = 7# times.
