# Question #18833

Sep 15, 2016

$9 : 04 : 40$ pm.

#### Explanation:

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.

$56 = {2}^{3} \times 7$
$40 = {2}^{3} \times 5$

We can see that the LCM requires at least three factors of $2$, one factor of $5$, and one factor of $7$. Thus

$L C M \left(40 , 56\right) = {2}^{3} \times 5 \times 7 = 280$.

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 $\frac{280}{56} = 5$ times, and the second will have flashed $\frac{280}{40} = 7$ times.