The generation time of a bacterium is the time it takes for it to double through cell division. If the generation time is 16 hours, each division results in a doubling of the bacterial population.
Let's calculate the total time it takes for 80 divisions to occur:
Number of divisions = 80
Generation time = 16 hours
Total time = Number of divisions × Generation time
\[ \text{Total time} = 80 \, \text{divisions} \times 16 \, \text{hours/division} \]
\[ \text{Total time} = 1280 \, \text{hours} \]
Now, to find the final time when the 80th division occurs, we add this total time to the initial time:
\[ \text{Final time} = \text{Initial time} + \text{Total time} \]
If the bacterium first divides at 4:00, we can substitute this into the equation:
\[ \text{Final time} = 4:00 \, \text{AM} + 1280 \, \text{hours} \]
To convert hours into a time format, we divide by 24 (since there are 24 hours in a day):
\[ \text{Final time} = 4:00 \, \text{AM} + \frac{1280 \, \text{hours}}{24 \, \text{hours/day}} \]
\[ \text{Final time} = 4:00 \, \text{AM} + 53 \, \text{days and } 8 \, \text{hours} \]
Therefore, the 80th division will occur 53 days and 8 hours after the bacterium first divides, and the final time will be 57:00 (5:00) AM.