Question
Vhich of the following rational numbers can be expressed as a repeating decimal: 7/8, 5/12, 17/25, or 31/40? (1 point)
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GPT-4o mini
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:
1. **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.
2. **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.
3. **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.
4. **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**.
Let's factor the denominators of each of the given fractions:
1. **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.
2. **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.
3. **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.
4. **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**.
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