- The multiples of 3 are \(\{3, 6, 9, 12, 15, \dots\}\).
- The multiples of 6 are \(\{6, 12, 18, 24, 30, \dots\}\).
- The common multiples are \(\{6, 12, \dots\}\).
- The smallest value, which is the least common multiple, is \(\boxed{6}\).
Explanation
- Problem Analysis and Definition of LCM
The problem asks for the least common multiple (LCM) of the numbers 3 and 6, using the provided lists of their multiples. The LCM of two integers is defined as the smallest positive integer that is divisible by both of the numbers.
From the table provided:
- The multiples of 3 are: {3, 6, 9, 12, 15, ...}
- The multiples of 6 are: {6, 12, 18, 24, 30, ...}
- Identifying Common Multiples and the LCM
To find the common multiples, we identify the numbers that appear in both lists.
- Multiples of 3: 3, 6, 9, 12, 15, ...
- Multiples of 6: 6, 12, 18, 24, 30, ...
The common multiples are 6, 12, ... and so on. The least common multiple (LCM) is the smallest positive integer in the set of common multiples. In this case, the smallest common multiple is 6.
- Final Answer
Since 6 is the smallest number that is a multiple of both 3 and 6, it is the least common multiple. Therefore, \(\text{lcm}(3, 6) = 6\). The final answer is \(\boxed{6}\).
Examples
Imagine you are setting up two different automatic timers in a smart home system. One light turns on every 3 minutes, and a separate security camera system checks in every 6 minutes. By finding the least common multiple (LCM) of these intervals, which is \(\text{lcm}(3, 6) = 6\), you determine that both the light and the security camera will coincide every 6 minutes. This helps in efficiently coordinating household automated tasks!