If the HCF and LCM of two consecutive (positive) even numbers be 2 and 84 respectively, then the sum of the numbers is

Its very simple question 26 is the correct answer.

we know that HCF×LCM=ist No.×2nd No.
let the one no.be x then the consecutive even no.=x+2
so,$x(x+2)=2×84=168$
$x^2 +2x-168=0$
splitting middle term we get
$x^2+14x-12x-168=0$
$x(x+14)-12(x-14)=0$
$x+14=0 or x-12=0$
we take $x-12=0$ becoz $x+14$ will give -ve value of x
so $x=12 $ & $x+2=14$
.'. $x+(x+2)=26$
so the sum of two consecutive even no.s with hcf $=2$ and lcm $=84$ is $26$.

Let the numbers be 2x and 2y
where x and y are prime to each
other.
LCM = 2xy
2xy = 84
xy = 42 = 6 × 7
Numbers are 12 and 14.
Hence Sum = 12 + 14 = 26