Even Integer Properties

Consider the following problem involving divisibility and properties of integers.

Let $a$, $b$, and $c$ be integers defined as follows:

• $a = 4x + 2$
• $b = 7y - 1$
• $c = 3z + 5$

For which of the following values can it be guaranteed that $a$, $b$, and $c$ are all even integers?