Let us assume x be the smaller of the two consecutive even positive integers

∴ Other integer = x + 2

It is also given in the question that, both the integers are larger than 5

∴ x > 5 (i)

Also, it is given in the question that sum off two integers is less than 23

∴ x + (x + 2) < 23

2x + 2 < 23

x > 10.5 (ii)

Thus, from (i) and (ii) we have x is an even number and it can take values 6, 8 and 10

Hence, possible pairs are (6, 8), (8, 10) and (10, 12)