1. Problem: Find the H.C.F of $6x$ and $3y$. 2. Factor each monomial into numeric primes and variable factors: $6x = 2\cdot 3\cdot x$ and $3y = 3\cdot y$. 3. Identify common factors: the numeric common factor is $3$ because $\gcd(6,3)=3$. 4. The variables $x$ and $y$ are distinct, so there is no common variable factor unless additional information is given. 5. Therefore the H.C.F is $3$. 6. Final answer: $\mathrm{H.C.F}(6x,3y)=3$.