Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.

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 ... (1)

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

*x*
+ (*x*
+ 2) < 23

⇒ 2*x*
+ 2 < 23

⇒ 2*x*
< 23 – 2

⇒ 2*x*
< 21

From
(1) and (2), we obtain 5 < *x*
< 10.5.

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).

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