A fraction in the form \( \frac{a}{b} \) (where \( a \) and \( b \) are integers and \( a < b \)) will convert to a repeating decimal if the denominator \( b \) has prime factors other than 2 and 5.
Let's check each option:
A. 18
- The prime factorization of 18 is \( 2 \times 3^2 \).
- Since it has a prime factor of 3, fractions with 18 as the denominator will convert to a repeating decimal.
B. 6
- The prime factorization of 6 is \( 2 \times 3 \).
- Since it has a prime factor of 3, fractions with 6 as the denominator will convert to a repeating decimal.
C. 11
- The prime factorization of 11 is \( 11 \) (which is prime).
- Since 11 is not 2 or 5, fractions with 11 as the denominator will convert to a repeating decimal.
D. 15
- The prime factorization of 15 is \( 3 \times 5 \).
- Since it has a prime factor of 3, fractions with 15 as the denominator will convert to a repeating decimal.
Since the question asks for the denominator that always converts to a repeating decimal, any option with a prime factor other than \( 2 \) or \( 5 \) is suitable.
The answer is:
- C. 11 is the only option that qualifies as it has no factors of 2 or 5 and is always a repeating decimal.
So the correct choice is: C. 11
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