Identify Given Data and Ratio Properties

Let the two numbers be represented in terms of their ratio $a : b = 6 : 7$. Since $6$ and $7$ are co-prime numbers (i.e., $\text{HCF}(6, 7) = 1$), the actual numbers are formed by multiplying each term of the ratio by their $\text{HCF} = 9$.

$\text{Ratio } a : b = 6 : 7, \quad \text{HCF} = 9$

Formulate the Numbers

The two numbers $A$ and $B$ are given by $A = a \times \text{HCF}$ and $B = b \times \text{HCF}$.

$A = 6 \times 9 = 54, \quad B = 7 \times 9 = 63$

Apply the Direct Formula for LCM

The Least Common Multiple (LCM) of two numbers expressed as a ratio $a : b$ with a given $\text{HCF}$ is computed as $\text{LCM} = \text{HCF} \times a \times b$.

$\text{LCM} = 9 \times 6 \times 7$

Calculate the Final Product

Multiply $9 \times 6 = 54$, then multiply $54 \times 7 = 378$.

$\text{LCM} = 378$