Answered

29. If \( x, y \), and \( z \) are positive integers such that \( 0 < x < y < z \) and \( x \) is even, \( y \) is odd, and \( z \) is prime, which of the following is a possible value of \( x + y + z \)?

A. 10
B. 15
C. 20
D. 25



Answer :

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].