Find all pairs of consecutive **even positive integers** both greater than 5 such that their sum is less than 23. How many such pairs exist?

Let $x$ be the smaller even integer. Then $x + 2$ is the next, with $x > 5$ (so smallest even $x = 6$) and $x + (x + 2) < 23 \Rightarrow x < 10.5$. Even $x$ with $5 < x < 10.5$ and $x + 2$ also even (automatic): $x \in \{6, 8, 10\}$ — **3 values**. Pairs: $(6, 8)$, $(8, 10)$, $(10, 12)$ — sums $14, 18, 22$, all $< 23$ ✓.