Numbers Finder by Sum and HCF

Input the sum and HCF of two numbers to calculate their actual values quickly.

Calculate Two Numbers Using Their Sum and HCF

Result

How to Calculate Two Numbers Using Their Sum and HCF

Suppose the two numbers are \( x \) and \( y \), and their sum is \( S \), with the highest common factor \( \text{HCF} \) given.

Steps

1. Understand the role of the HCF

Assume the two numbers can be expressed as: \( x = k \cdot a, \quad y = k \cdot b \) Here, \( k \) is the HCF, and \( a \) and \( b \) are two coprime integers (i.e., \( \text{HCF}(a, b) = 1 \)). The sum of the numbers is: \( x + y = k \cdot (a + b) = S \)

2. Calculate the proportional coefficient \( k \)

Using the sum \( S \) and the HCF, the proportional coefficient is given by: \( k = \text{HCF} \)

3. Find the coprime integers \( a \) and \( b \)

Since \( S = k \cdot (a + b) \), we can deduce: \( a + b = \frac{S}{\text{HCF}} \) List all pairs of integers that sum to \( a + b \) and check which ones are coprime.

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

Multiply \( k \) by \( a \) and \( b \) to find \( x \) and \( y \): \( x = k \times a \) \( y = k \times b \)

How to Find Coprime Pairs

List possible pairs: For a given \( a + b \), list all possible integer pairs \( (a, b) \). For example, when \( a + b = 6 \) , the possible pairs are \( (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) \) .
Check coprimality: Verify that the greatest common divisor (GCD) of each pair is 1. For example, \( (1, 5) \) is coprime, but \( (2, 4) \) is not because their GCD is 2.

Examples

Example 1: The sum of two numbers is 135 and their HCF is 27, what are the numbers?

Solution:

1. Determine the proportional coefficient \( k \):

\( k = HCF = 27 \)

2. Calculate \( a + b \):

\( a + b = \frac{135}{27} = 5 \)

3. Find coprime pairs:

For \( a + b = 5 \), the coprime pairs are \( (1, 4) \) and \( (2, 3) \).

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

\( x_1 = 27 \times 1 = 27 \)

\( y_1 = 27 \times 4 = 108 \)

\( x_2 = 27 \times 2 = 54 \)

\( y_2 = 27 \times 3 = 81 \)

Result: The two numbers are \( (27, 108) \) or \( (54, 81) \).

Example 1: The sum of two numbers is 629 and their HCF is 37, what are the numbers?

Solution:

1. Determine the proportional coefficient \( k \):

\( k = HCF = 37 \)

2. Calculate \( a + b \):

\( a + b = \frac{629}{37} = 17 \)

3. Find coprime pairs:

For \( a + b = 17 \), the coprime pairs are:

\( (1, 16) \)

\( (2, 15) \)

\( (3, 14) \)

\( (4, 13) \)

\( (5, 12) \)

\( (6, 11) \)

\( (7, 10) \)

\( (8, 9) \)

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

\( x_1 = 37 \times 1 = 37 \)

\( y_1 = 37 \times 16 = 592 \)

\( x_2 = 37 \times 2 = 74 \)

\( y_2 = 37 \times 15 = 555 \)

…

\( x_8 = 37 \times 8 = 296 \)

\( y_8 = 37 \times 9 = 333 \)

Result: Possible pairs are (37, 592), (74, 555), (111, 518), (148, 481), (185, 444), (222, 407), (259, 370), (296, 333).