To determine which of the given rational numbers can be expressed as a repeating decimal, we need to look at their denominators in simplest form. A rational number can be expressed as a repeating decimal if, after simplifying the fraction, its denominator has prime factors other than 2 and 5.
Let's analyze each fraction one by one:
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\( \frac{7}{8} \):
- The denominator \( 8 \) factors as \( 2^3 \).
- Since it only contains the prime factor 2, \( \frac{7}{8} \) can be expressed as a terminating decimal, specifically \( 0.875 \).
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\( \frac{5}{12} \):
- The denominator \( 12 \) factors as \( 2^2 \times 3 \).
- Since it contains the prime factor 3 (in addition to 2), \( \frac{5}{12} \) can be expressed as a repeating decimal.
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\( \frac{17}{25} \):
- The denominator \( 25 \) is \( 5^2 \).
- Since it only contains the prime factor 5, \( \frac{17}{25} \) can be expressed as a terminating decimal, specifically \( 0.68 \).
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\( \frac{31}{40} \):
- The denominator \( 40 \) factors as \( 2^3 \times 5 \).
- Since it only contains the prime factors 2 and 5, \( \frac{31}{40} \) can be expressed as a terminating decimal, specifically \( 0.775 \).
Based on this analysis, the only rational number from the list that can be expressed as a repeating decimal is \( \frac{5}{12} \).