Input the product and least common multiple (LCM) of two numbers to instantly determine the numbers.
To find two numbers \( x \) and \( y \) when their product (\( P \)) and least common multiple (LCM) are given, we rely on the relationship between the product, LCM, and greatest common divisor (GCD).
Using the fundamental relationship: \( x \times y = \text{LCM}(x, y) \times \text{GCD}(x, y) \) Rearrange to solve for GCD: \( \text{GCD}(x, y) = \frac{P}{\text{LCM}} \)
Represent \( x = \text{GCD} \cdot a \) and \( y = \text{GCD} \cdot b \), where \( a \) and \( b \) are coprime (mutually prime).
From \( x \cdot y = P \), we derive \( a \cdot b = \frac{P}{\text{GCD}^2} \).
Factorize \( a \cdot b \) to find all possible pairs \( (a, b) \). Select pairs where \( a \) and \( b \) are coprime (no common factors other than 1).
For each coprime pair \( (a, b) \), compute: \( x = \text{GCD} \times a \) \( y = \text{GCD} \times b \)
Solution:
1. Compute GCD:
\( \text{GCD} = \frac{2208}{552} = 4 \)
2. Solve for \( a \cdot b \):
\( a \times b = \frac{2208}{4^2} = 138 \)
3. Find coprime factor pairs of 138:
Factorize \( 138 \): \( (1, 138), (2, 69), (3, 46), (6, 23) \) .
These factor pairs all satisfy the coprime condition.
4. Compute \( x \) and \( y \):
\( x_1 = 4 \times 1 = 4 \)
\( y_1 = 4 \times 138 = 552 \)
\( x_2 = 4 \times 2 = 8 \)
\( y_2 = 4 \times 69 = 276 \)
\( x_3 = 4 \times 3 = 12 \)
\( y_3 = 4 \times 46 = 184 \)
\( x_4 = 4 \times 6 = 24 \)
\( y_4 = 4 \times 23 = 92 \)
Result: The two numbers are (552, 4), (276, 8), (184, 12) or (92, 24).
Solution:
1. Compute GCD:
\( \text{GCD} = \frac{2560}{320} = 8 \)
2. Solve for \( a \cdot b \):
\( a \times b = \frac{2560}{8^2} = 40 \)
3. Find coprime factor pairs of 40:
Factorize \( 40 \): \( (1, 40), (2, 20), (4, 10), (5, 8)\)
Identify coprime pairs: \( (1, 40)\) and \( (5, 8)\)
4. Compute \( x \) and \( y \):
\( x_1 = 8 \times 1 = 8 \)
\( y_1 = 8 \times 40 = 320 \)
\( x_2 = 8 \times 5 = 40 \)
\( y_2 = 8 \times 8 = 64 \)
Result: The two numbers are (8, 320) or (40, 64).