Question

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

Answer 1

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 
⇒ \(\frac{2x}{2} <\frac{ 21}{2}\)
⇒ 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).