If the dark green rods (6 units) and the brown rods (8 units) are each used to form the shortest possible one-color train of matching length, how many dark green rods and how many brown rods will be required? What information about common factors or common multiples does this provide?
The shortest possible one-color train of matching length can be formed by finding the least common multiple (LCM) of the two rod lengths.
To find the LCM of 6 and 8, we can list their multiples:
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60...
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80...
The number 24 appears in both lists, so the LCM of 6 and 8 is 24.
Therefore, to form the shortest possible one-color train of matching length, we will need 4 dark green rods (6 units each) and 3 brown rods (8 units each).
This information provides insight into the common factors and common multiples of the two rod lengths. The LCM of two numbers is the smallest number that is divisible by both of them. In this case, 24 is the smallest length at which both the dark green rods (6 units) and brown rods (8 units) align. Additionally, we can also see that 24 can be factored into 6 * 4 and 8 * 3, indicating that the rod lengths have common factors.