Question

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

• A. $(6,8),(8,10),(10,12)$
• B. $(3,5),(7,9),(8,10)$
• C. $(3,5),(5,7),(7,9)$
• D. $(4,6),(8,10),(10,12)$

Answer 1

Answer: (A)

$(6,8),(8,10),(10,12)$

Let the smaller of the two consecutive even numbers be $x$
Therefore the larger number will be $x+2$.

It is given that their sum is less than $23$
$x+x+2<23$
$2x<21$
$x<\frac{21}{2}$
$x<\frac{20+1}{2}$
$x<10+\frac{1}{2}$

It is given that $x$ is an integer
Therefore we can neglect the fractional part of $x$
$x<10$
$(x+2)<12$ ..... Adding $2$ on both sides

Hence the smaller of the consecutive number is less than $10$, while the larger is less than $12$ ...(i)
Also both the consecutive numbers are larger than $5$ ...(ii)
Hence from statements i and ii the answer is A.