Numbers Finder by HCF and LCM

Input the HCF and LCM of two numbers and calculate the numbers quickly and accurately.

Calculate Two Numbers Based on Their HCF and LCM

Result

How to Calculate Two Numbers Based on Their HCF and LCM

Assume the two numbers are \( x \) and \( y \), with their highest common factor (\( \text{HCF} \)) and least common multiple (\( \text{LCM} \)) provided.

Step-by-Step Derivation

1. Relationship Between HCF and LCM

According to the basic properties of numbers, we know: \( x \times y = \text{HCF} \times \text{LCM} \) Denote the product of \( x \) and \( y \) as \( P \): \( P = \text{HCF} \times \text{LCM} \)

2. Assume Proportional Coefficients

Let \( x = \text{HCF} \times a \) and \( y = \text{HCF} \times b \), where \( a \) and \( b \) are coprime integers (i.e., their greatest common divisor is 1). From this, we derive: \( a \times b = \frac{P}{\text{HCF}^2} = \frac{\text{HCF} \times \text{LCM}}{\text{HCF}^2} = \frac{\text{LCM}}{\text{HCF}} \)

3. Find Coprime Integer Pair \( a \) and \( b \)

Based on the calculated value \( a \times b = \frac{\text{LCM}}{\text{HCF}} \), determine all pairs of integers that satisfy the equation and are coprime.

4. Calculate \( x \) and \( y \)

Multiply the HCF by the determined values of \( a \) and \( b \) to compute \( x \) and \( y \): \( x = \text{HCF} \times a \) \( y = \text{HCF} \times b \)

Examples

Example 1: Find two numbers whose HCF is 179 and LCM is 2685.

Solution:

1. Compute the product \( P \):

\( P = \text{HCF} \times \text{LCM} = 179 \times 2685 = 480015 \)

2. Compute \( a \times b \):

\( a \times b = \frac{\text{LCM}}{\text{HCF}} = \frac{2685}{179} = 15 \)

3. Find coprime pairs for \( a \times b = 15 \):

Factorize 15 into pairs: (1, 15) and (3, 5). Both are coprime.

4. Calculate \( x \) and \( y \):

\( x_1 = 179 \times 1 = 179 \)

\( y_1 = 179 \times 15 = 2685 \)

\( x_2 = 179 \times 3 = 537 \)

\( y_2 = 179 \times 5 = 895 \)

Result: The numbers are (179, 2685) or (537, 895).

Example 2: Find two numbers whose HCF is 11 and LCM is 7700.

Solution:

1. Compute the product \( P \):

\( P = \text{HCF} \times \text{LCM} = 11 \times 7700 = 84700 \)

2. Compute \( a \times b \):

\( a \times b = \frac{\text{LCM}}{\text{HCF}} = \frac{7700}{11} = 700 \)

3. Find coprime pairs for \( a \times b = 700 \):

Factorize 700 into pairs:

(1, 700)
(2, 350)
(4, 175)
(5, 140)
(7, 100)
(10, 70)
(14, 50)
(20, 35)
(25, 28)

Coprime pairs are (1, 700), (4, 175), (7, 100), (25, 28).

4. Calculate \( x \) and \( y \):

\( x_1 = 11 \times 1 = 11 \)

\( y_1 = 11 \times 700 = 7700 \)

\( x_2 = 11 \times 4 = 44 \)

\( y_2 = 11 \times 175 = 1925 \)

\( x_3 = 11 \times 7 = 77 \)

\( y_3 = 11 \times 100 = 1100 \)

\( x_4 = 11 \times 25 = 275 \)

\( y_4 = 11 \times 28 = 308 \)

Result: The numbers are (11, 7700), (44, 1925), (77, 1100) or (275, 308).