Step 1: Understand what is being asked.
We are given two numbers, 50 and 20, and asked to find the product of their H.C.F. and L.C.M. Instead of using the shortcut property directly, let us find the H.C.F. and L.C.M. separately using the division method, and then multiply them.
Step 2: Find the H.C.F. using Euclid's division algorithm.
Divide the larger number by the smaller number and keep dividing the divisor by the remainder until the remainder becomes 0.
Divide 50 by 20: $50 = 20 \times 2 + 10$
Divide 20 by the remainder 10: $20 = 10 \times 2 + 0$
Since the remainder is now 0, the last divisor is the H.C.F.
\[ \text{H.C.F.}(50, 20) = 10 \]
Step 3: Find the L.C.M. using the H.C.F.
For any two numbers $a$ and $b$, the product of the numbers equals the product of their H.C.F. and L.C.M. This gives a formula to get the L.C.M. once the H.C.F. is known:
\[ \text{L.C.M.} = \frac{a \times b}{\text{H.C.F.}} \]
Substitute the values:
\[ \text{L.C.M.}(50, 20) = \frac{50 \times 20}{10} = \frac{1000}{10} = 100 \]
Step 4: Multiply the H.C.F. and L.C.M.
\[ \text{H.C.F.} \times \text{L.C.M.} = 10 \times 100 = 1000 \]
This confirms the required product without ever multiplying 50 and 20 directly.
Final Answer:
The product of the H.C.F. and L.C.M. of 50 and 20 is 1000.
\[ \boxed{1000} \]