Question

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

• A. $(4,6),(6,8),(8,10),(10,12)$
• B. $(6,8),(8,10),(10,12)$
• C. $(6,8),(8,10),(10,12),(12,14)$
• D. $(8,10),(10,12)$

Answer 1

Answer: (B)

Let numbers are $2x$ and $2x+2$
Then, according to the question,
$2x>5\Rightarrow x>\frac{5}{2}$
$\text{ and }2x+2>5\Rightarrow 2x>5-2$
$\Rightarrow 2x>3\Rightarrow x>3/2$
$\text{ and }2x+2x+2<23\Rightarrow 4x<23-2$
$\Rightarrow 4x<21$
$\Rightarrow x<\frac{21}{4}$
Now, plotting all these values on number line.

From above graph, it is clear that $x\in \left(\frac{5}{2},\frac{21}{4}\right)$ in which integer values are $x=3,4,5$
When $x=3$, pair is $(2\times 3,2\times 3+2)=(6,8)$
When $x=4$, pair is $(2\times 4,2\times 4+2)=(8,10)$
When $x=5$, pair is $(2\times 5,2\times 5+2)=(10,12)$
$\therefore$ Required pairs are $(6,8),(8,10),(10,12)$.