If the mean (average) of five consecutive integers is $21,$ what is the smallest of the five integers?
The mean of five consecutive integers is equal to the number in the middle. Since the numbers have a mean of $21,$ if we were to distribute the quantities equally, we would have $21,$ $21,$ $21,$ $21,$ and $21.$ Since the numbers are consecutive, the second number is $1$ less than the $21$ in the middle, while the fourth number is $1$ more than the $21$ in the middle. Similarly, the first number is $2$ less than the $21$ in the middle, while the fifth number is $2$ more than the $21$ in the middle.

Thus, the numbers are $21-2,$ $21-1,$ $21,$ $21+1,$ $21+2.$ The smallest of $5$ consecutive integers having a mean of $21,$ is $\boxed{19}.$