1. **State the problem:** We want to determine if the sum of three consecutive integers $a$, $b$, and $c$ is always even. 2. **Recall the definition of consecutive integers:** Consecutive integers differ by 1, so if $a$ is an integer, then $b = a + 1$ and $c = a + 2$. 3. **Express the sum:** $$a + b + c = a + (a + 1) + (a + 2)$$ 4. **Simplify the sum:** $$a + (a + 1) + (a + 2) = 3a + 3 = 3(a + 1)$$ 5. **Analyze parity:** Since $3(a + 1)$ is a multiple of 3, but not necessarily a multiple of 2, the sum is not always even. 6. **Counterexample:** If $a = 1$, then $a + b + c = 1 + 2 + 3 = 6$ (even). If $a = 2$, then $a + b + c = 2 + 3 + 4 = 9$ (odd). 7. **Conclusion:** The statement "for all integers $a$, $b$, and $c$, if they are consecutive, then $a + b + c$ is even" is **false** because the sum can be odd or even depending on $a$.