To find a possible value of \( x + y + z \) given the conditions, let's start by understanding the constraints we need to satisfy:
1. \( x \) is a positive even integer.
2. \( y \) is a positive odd integer.
3. \( z \) is a positive prime number.
4. \( 0 < x < y < z \).
Starting with the smallest positive even integer, we select:
[tex]\[ x = 2 \][/tex]
This choice is motivated by \( x \) needing to be not only even but also the smallest such integer greater than 0.
Next, to satisfy \( y \) being an odd integer and greater than \( x \), we consider the smallest odd integer greater than 2:
[tex]\[ y = 3 \][/tex]
Finally, \( z \) needs to be the smallest prime number that is greater than \( y \):
[tex]\[ z = 5 \][/tex]
Now, we need to compute the sum \( x + y + z \):
[tex]\[ x + y + z = 2 + 3 + 5 = 10 \][/tex]
Therefore, a possible value of [tex]\( x + y + z \)[/tex] is [tex]\(\boxed{10}\)[/tex].
