To determine which number is a multiple of both 6 and 8, we need to find the Least Common Multiple (LCM) of these two numbers. The LCM of two numbers is the smallest number that is a multiple of both.
Here's a detailed step-by-step solution for this problem:
1. Identify the numbers given in the options and the numbers to find the LCM for:
- Numbers to find LCM for: 6 and 8
- Given options: 18, 2, 3, 24
2. Break down the factors of the numbers 6 and 8:
- 6 factors into [tex]\(2 \cdot 3\)[/tex]
- 8 factors into [tex]\(2^3\)[/tex]
3. Determine the highest power of each prime factor that appears in these factorizations:
- The highest power of 2 between the two numbers is [tex]\(2^3\)[/tex] (from 8)
- The highest power of 3 is [tex]\(3^1\)[/tex] (from 6)
4. Calculate the LCM using these highest powers:
[tex]\[
\text{LCM} = 2^3 \cdot 3 = 8 \cdot 3 = 24
\][/tex]
5. Check which of the given options matches the LCM:
- 18: Not a multiple of both 6 and 8
- 2: Not a multiple of both 6 and 8
- 3: Not a multiple of both 6 and 8
- 24: A multiple of both 6 and 8
Therefore, the correct answer is:
24
This means 24 is the smallest number that is a multiple of both 6 and 8, making it the correct option from the given choices.
