Find the number of pairs of consecutive even positive integers, both of which are greater than 5 such that their sum is less than 23.
6
(8, 10)
(10, 12)
Let x be the smaller of the two consecutive even positive integers. Then, other integer is x+2
Since, both the integers are greater than 5,
x>5 ------(1)
Also, the sum of two integers is less than 23.
⇒ x+(x+2)<23
⇒ 2x<23–2
⇒ 2x<21
⇒ 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, 10
Thus, required possible pairs are (6, 8), (8, 10) and (10,12).