Find all pairs of consecutive even positive integers,

Let x be the smaller of the two consecutive even positive integers. Then, the other integer is x + 2.

Since both the integers are larger than 5,

$x>5 \ldots$ (1)

Also, the sum of the two integers is less than 23 .

$x+(x+2)<23$

$\Rightarrow 2 x+2<23$

$\Rightarrow 2 x<23-2$

$\Rightarrow 2 x<21$

$\Rightarrow x<\frac{21}{2}$

$\Rightarrow x<10.5$  $\ldots$ (2)

From (1) and (2), we obtain $5

Since $x$ is an even number, $x$ can take the values, 6,8 , and 10 .

Thus, the required possible pairs are $(6,8),(8,10)$, and $(10,12)$.