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
2x + 2 < 23
⇒ 2x < 23 – 2
⇒ 2x < 21
⇒x<21/2
⇒x<10.5 .......(2)
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).