Question

The product of the H.C.F. and L.C.M. of two numbers 50 and 20 is :

• A. 100
• B. 1000
• C. 50
• D. 20

Hint:

For any two positive numbers, you do not need to calculate their H.C.F. and L.C.M. separately to find their product.
Directly multiplying the two given numbers is much faster and saves valuable exam time.
Note that this property only holds true for two numbers and cannot be directly generalized to three or more numbers.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks us to find the product of the Highest Common Factor (H.C.F.) and the Least Common Multiple (L.C.M.) of two given numbers, namely 50 and 20.

Step 2: Key Formula or Approach:
We can use the fundamental property of real numbers relating the H.C.F. and L.C.M. of two positive integers \(a\) and \(b\):
\[ \text{H.C.F.}(a, b) \times \text{L.C.M.}(a, b) = a \times b \]
This relationship states that the product of the H.C.F. and L.C.M. of any two numbers is always equal to the product of the two numbers themselves.

Step 3: Detailed Explanation:
1. Let the two given numbers be \(a = 50\) and \(b = 20\).
2. According to the formula, the product of their H.C.F. and L.C.M. is given by the product of the numbers:
\[ \text{Product} = a \times b \]
3. Substituting the values of \(a\) and \(b\):
\[ \text{Product} = 50 \times 20 \]
\[ \text{Product} = 1000 \]
4. Alternatively, we can find the individual values of H.C.F. and L.C.M. to verify the relationship.
Prime factorization of 50:
\[ 50 = 2 \times 5^2 \]
Prime factorization of 20:
\[ 20 = 2^2 \times 5 \]
H.C.F. is the product of the lowest power of common prime factors:
\[ \text{H.C.F.}(50, 20) = 2^1 \times 5^1 = 10 \]
L.C.M. is the product of the highest power of all prime factors involved:
\[ \text{L.C.M.}(50, 20) = 2^2 \times 5^2 = 4 \times 25 = 100 \]
Multiplying the calculated H.C.F. and L.C.M. gives:
\[ \text{H.C.F.} \times \text{L.C.M.} = 10 \times 100 = 1000 \]
Both methods yield the same result of 1000.

Step 4: Final Answer:
Therefore, the product of the H.C.F. and L.C.M. of the numbers 50 and 20 is 1000, which corresponds to option (B).