The average of three different positive integers is 8. what is the largest integer that could be one of them.

Supposing that the integers are x, y and z, and that their average is 8, then

x+y+z
= 8, or x + y + z = 24.
    3

Let's experiment.    Making up a table, we get:

x   y   z   sum (sum MUST be 24)
5  9  10    24
4  9   11    24
3  10  11   24
2   11 11    24
1   11  12   24
1   10  13   24
1    1   22   24

Can't choose integers smaller than 1, so it appears that 22 is the largest possible integer that could be one of x, y and z.