Numbers Finder by Difference and LCM

Input the difference and LCM of two numbers to quickly calculate the numbers.

How to Calculate Numbers Based on Their Difference and LCM

Let's assume two numbers \( x \) and \( y \), where: Difference: \( x - y = D \) Least Common Multiple (LCM): \( \text{LCM}(x, y) \)

Follow these steps to determine \( x \) and \( y \):

List all factors of the LCM: Identify all factors of the given \( \text{LCM} \). These factors represent possible values for \( x \) and \( y \).
Filter pairs with a difference of \( D \): From the list of factors, select pairs \( (x, y) \) where \( x - y = D \). Start by assigning any factor as \( y \), then compute \( x = y + D \). If \( x \) is also in the factor list, you've found a valid pair \( (x, y) \).
Verify the LCM: For each candidate pair \( (x, y) \), calculate their LCM and check if it matches the given \( \text{LCM} \). If the LCM matches, the pair is the solution.

Solution:

1. List all factors:

Factors of 495: 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495.

2. Find pairs with a difference of 10:

\( (5, 15) \)

\( (45, 55) \)

3. Verify the LCM:

\( \text{LCM}(5, 15) = 15 \), Does not match.

\( \text{LCM}(45, 55) = 495 \), Matches.

Result: The two numbers are 55 and 45.

Solution:

1. List all factors:

Factors of 1008: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 36, 42, 48, 56, 63, 72, 84, 112, 126, 144, 168, 252, 336, 504, 1008.

2. Find pairs with a difference of 81:

\( (3, 84) \)

\( (63, 144) \)

3. Verify the LCM:

\( \text{LCM}(3, 84) = 84 \), Does not match.

\( \text{LCM}(63, 144) = 1008 \), Matches.

Result: The two numbers are 144 and 63.