The average (arithmetic mean) of the positive integers x, y, and z is 3
We can write: $\frac{x+y+z}{3}=3$

Multiply both sides of the equation by $3$ to get: $x+y+z=9$

At this point we need to find the greatest possible value of $z$
Important: Keep in mind that the three numbers are DIFFERENT POSITIVE INTEGERS

Since we know that $x+y+z=9$, we can MAXIMIZE the value of $z$ by MINIMIZING the values of $x$ and $y$
We know that $x$ is the smallest value.
Since $x$ must be a positive integer, the smallest possible value of $x$ is $1$
Since $y$ must be different from $x$, the smallest possible value of $y$ is $2$
At this point we have maximized the value of $z$

If $x=1$ and $y=2$, then $z=6$

Answer: B

Cheers,
Brent