Property 1: The product of LCM and HCF of any two given natural numbers is equivalent to the product of the given numbers.
$LCM \times HCF = \text{Product of the Numbers}$
Suppose $A$ and $B$ are two numbers, then.
$LCM (A \& B) \times HCF (A \& B) = A \times B$
Property 2: HCF of co-prime numbers is $1.$ Therefore LCM of given co-prime numbers is equal to the product of the numbers.
LCM of Co-prime Numbers $=$ Product Of The Numbers
Property 3: H.C.F. and L.C.M. of Fractions
LCM of fractions $= \dfrac {\text{LCM of numerators}} {\text{HCF of denominators}}$
HCF of fractions $= \dfrac{\text{HCF of numerators}} {\text{LCM of denominators}}$
Given that $HCF$ is $23$, so we can let the two numbers are $A = 23x\:\:\&\:\:B = 23y,$ where $x$ and $y$ are co-prime to each other.
$LCM(A,B) \times HCF(A,B) = 23 x \times 23 y$
$LCM(A , B) \times 23 = 23 x \times 23 y$
$LCM(A , B) = 23\times x \times y$
Given that other two factors of LCM is $13\: \&\: 14$
$\therefore x = 13,y=14$
So, numbers are $A = 23x = 23 \times 13 = 299$ and $B = 23y = 23 \times 14 = 322$
Therefore, $max(A,B) = max(299,322) = 322$
So, the correct answer is $322$.