To determine which of the given rational numbers can be expressed as a repeating decimal, we need to understand the criteria for a rational number to have a terminating decimal or a repeating decimal.
A rational number can be expressed as a fraction \(\frac{p}{q}\) (where \(p\) and \(q\) are integers and \(q \neq 0\)), and its decimal representation is repeating if the denominator \(q\) (when reduced to lowest terms) has prime factors other than 2 and 5. If \(q\) only has these prime factors, the decimal is terminating.
Let's examine each of the given numbers:
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78: This can be expressed as \(\frac{78}{1}\). The denominator \(1\) has no prime factors other than 2 and 5. Therefore, 78 has a terminating decimal.
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512: This can be expressed as \(\frac{512}{1}\). The denominator \(1\) again has no prime factors other than 2 and 5. Thus, 512 also has a terminating decimal.
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1725: This can be expressed as \(\frac{1725}{1}\). The prime factorization of 1725 is \(3 \times 5^2 \times 23\). The denominator \(1\) contributes no additional prime factors; however, since there is a prime factor of 3 in 1725, it can be concluded that \(\frac{1725}{1}\) eventually results in a repeating decimal.
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3140: This can be expressed as \(\frac{3140}{1}\). The prime factorization of 3140 is \(2 \times 5 \times 314\). The prime factorization of 314 (which is co-prime with 2 and 5 because it is \(2 \times 157\)) suggests the same argument as with 1725, which results in a repeating decimal here as well.
Thus summarizing the above:
- 78: Terminating decimal
- 512: Terminating decimal
- 1725: Repeating decimal
- 3140: Repeating decimal
From the above analyses, 1725 and 3140 are the rational numbers that can be expressed as a repeating decimal.
3 answers