A rational number can be expressed as a repeating decimal if its denominator (when the fraction is in simplest form) has any prime factors other than 2 or 5.
Let's analyze each of the fractions given:
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7/8: The denominator is 8, which is \(2^3\). Since it only has the prime factor 2, \( \frac{7}{8} \) can be expressed as a terminating decimal, \( 0.875 \).
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5/12: The denominator is 12, which factors into \(2^2 \times 3\). Since it has the prime factor 3 (which is not 2 or 5), \( \frac{5}{12} \) can be expressed as a repeating decimal.
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17/25: The denominator is 25, which is \(5^2\). Since it only has the prime factor 5, \( \frac{17}{25} \) can be expressed as a terminating decimal, \( 0.68 \).
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31/40: The denominator is 40, which factors into \(2^3 \times 5\). Since it only has the prime factors 2 and 5, \( \frac{31}{40} \) can also be expressed as a terminating decimal, \( 0.775 \).
Based on this analysis, the only rational number from the list that can be expressed as a repeating decimal is 5/12.
3 answers