To find the least common denominator (LCD) of the fractions \( \frac{1}{6} \) and \( \frac{5}{8} \), we need to determine the least common multiple (LCM) of the denominators 6 and 8.
First, we find the prime factorizations of the denominators:
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The prime factorization of \( 6 \) is: \[ 6 = 2 \times 3 \]
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The prime factorization of \( 8 \) is: \[ 8 = 2^3 \]
Next, we take the highest power of each prime factor present in these factorizations:
- For the prime \( 2 \): the highest power is \( 2^3 \) (from 8).
- For the prime \( 3 \): the highest power is \( 3^1 \) (from 6).
Now, we find the LCM by multiplying these highest powers together:
\[ \text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24 \]
Thus, the least common denominator of \( \frac{1}{6} \) and \( \frac{5}{8} \) is \( \boxed{24} \).