For a us census clock based on the 2000 census, the birth light would flash every 10 seconds, the death light every 16 seconds, the immigration light every 81 seconds, and the emigration light every 900 seconds, to indicate gains and losses in population.

a. if the birth and death lights flashed at the same time, how many seconds would pass before they flashed together again?

show work

80

Step-by-step explanation:

Let's pretend we start a stopwatch when the birth and death light flash together.  The birth light will flash at 10 seconds, 20 seconds, 30 seconds, etc.  The death light will flash at 16 seconds, 32 seconds, 48 seconds, etc.

So the trick here is to find the least common multiple (LCM) of 10 and 16.  To do that, we write the prime factorization of each:

10 = 2×5

16 = 2⁴

The LCM must include all these prime factors without any duplicates.  So the LCM = 2⁴×5 = 80.  We can check our answer by dividing:

80 / 10 = 8

80 / 16 = 5

So 80 is indeed a multiple of both 10 and 16.

the mistake in the pattern is that :

b. the decimal point was moved one too many place values each time

the correct pattern could be made by :

b. moving the decimal point to the right the same numbers of place values as the exponents