To find the lowest common denominator (LCD) of the given fractions [tex]\(\frac{1}{2}, \frac{1}{3}, \frac{1}{5}, \frac{1}{6},\)[/tex] and [tex]\(\frac{1}{8}\)[/tex], we need to determine the least common multiple (LCM) of their denominators: 2, 3, 5, 6, and 8.
Here's a step-by-step method to find the LCM:
1. List the denominators: 2, 3, 5, 6, 8.
2. Prime factorize each denominator:
- [tex]\(2 = 2\)[/tex]
- [tex]\(3 = 3\)[/tex]
- [tex]\(5 = 5\)[/tex]
- [tex]\(6 = 2 \times 3\)[/tex]
- [tex]\(8 = 2^3\)[/tex]
3. Identify the highest power of each prime factor present in any of the denominators:
- The prime factor 2 has its highest power as [tex]\(2^3\)[/tex] (from 8).
- The prime factor 3 has its highest power as [tex]\(3\)[/tex] (from both 3 and 6).
- The prime factor 5 has its highest power as [tex]\(5\)[/tex].
4. Multiply these highest powers together to find the LCM:
- [tex]\(2^3 = 8\)[/tex]
- [tex]\(3 = 3\)[/tex]
- [tex]\(5 = 5\)[/tex]
- Now, multiply these values together: [tex]\(8 \times 3 \times 5 = 120\)[/tex]
Therefore, the least common multiple of the denominators 2, 3, 5, 6, and 8 is 120.
Thus, the lowest common denominator for the fractions [tex]\(\frac{1}{2}, \frac{1}{3}, \frac{1}{5}, \frac{1}{6},\)[/tex] and [tex]\(\frac{1}{8}\)[/tex] is [tex]\(\boxed{120}\)[/tex].
