A rational number can be expressed as a repeating decimal if its denominator (in lowest terms) has prime factors other than 2 and 5.
Let's analyze each of the given fractions:
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7/8: The denominator is 8, which is \(2^3\). This means it can be expressed as a terminating decimal.
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5/12: The denominator is 12, which factors as \(2^2 \times 3\). The presence of 3 means it cannot be expressed as a terminating decimal, so it is a repeating decimal.
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17/25: The denominator is 25, which is \(5^2\). This means it can be expressed as a terminating decimal.
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31/40: The denominator is 40, which factors as \(2^3 \times 5\). This means it can also be expressed as a terminating decimal.
From this analysis, the only fraction that can be expressed as a repeating decimal is 5/12.