A rational number can be expressed as a repeating decimal if, when expressed in simplest form, its denominator has primes other than 2 and 5 as factors.
Let's factor the denominators of each of the given fractions:
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7/8: The denominator is 8, which is \(2^3\). This has only the prime factor 2. So, it can be expressed as a terminating decimal.
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5/12: The denominator is 12, which factors into \(2^2 \times 3\). Since it has a prime factor of 3, it can be expressed as a repeating decimal.
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17/25: The denominator is 25, which is \(5^2\). This has only the prime factor 5. So, it can be expressed as a terminating decimal.
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31/40: The denominator is 40, which factors into \(2^3 \times 5\). This has only the prime factors 2 and 5. So, it can be expressed as a terminating decimal.
Among the options, the only rational number that can be expressed as a repeating decimal is 5/12.
3 answers