Input the sum and LCM of two numbers to quickly determine the two numbers.
Given two numbers \( x \) and \( y \), where: Sum: \( x + y = S \) Least Common Multiple (LCM): \( \text{LCM}(x, y) \)
Follow these steps to find the two numbers:
Solution:
1. List all factors:
Factors of 420: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420.
2. Filter pairs (sum = 144):
Potential pairs: \( (4, 140) \), \( (60, 84) \).
3. Verify the LCM:
\( \text{LCM}(4, 140) = 140 \), Does not match.
\( \text{LCM}(60, 84) = 420 \), Matches.
Result: The two numbers are 60 and 84.
Solution:
1. List all factors:
Factors of 960: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 160, 192, 240, 320, 480, 960.
2. Filter pairs (sum = 272):
Potential pairs: \( (32, 240) \)、\( (80, 192) \).
3. Verify the LCM:
\( \text{LCM}(32, 240) = 480 \), Does not match.
\( \text{LCM}(80, 192) = 960 \), Matches.
Result: The two numbers are 80 and 192.
Solution:
1. List all factors:
Factors of 21879: 1, 3, 9, 11, 13, 17, 33, 39, 51, 99, 117, 143, 153, 187, 221, 429, 561, 663, 1287, 1683, 1989, 2431, 7293, 21879.
2. Filter pairs (sum = 2000):
Potential pair: \( (11, 1989) \).
3. Verify the LCM:
\( \text{LCM}(11, 1989) = 21879 \), Matches.
Result: The two numbers are 11 and 1989.