Question

Maria rewrites a fraction less than 1 as a decimal.
The numerator is a whole number greater than 0. For which denominator will the fraction always convert to a repeating decimal?

A. 18
B. 6
C. 11
D. 15

Answer 1

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**