To determine which of the given rational numbers can be expressed as a repeating decimal, we need to check their denominators.
For a rational number to have a repeating decimal representation, its denominator must contain at least one prime factor that is not a factor of 10.
The prime factorization of the denominators are as follows:
7/8: The denominator, 8, has a prime factorization of $2^3$.
5/12: The denominator, 12, has a prime factorization of $2^2 \cdot 3$.
17/25: The denominator, 25, has a prime factorization of $5^2$.
31/40: The denominator, 40, has a prime factorization of $2^3 \cdot 5$.
Among the given rational numbers, only 5/12 and 31/40 have denominators that contain prime factors other than 2 or 5. Therefore, 5/12 and 31/40 can be expressed as repeating decimals.
Which of the following rational numbers can be expressed as a repeating decimal: 7/8, 5/12, 17/25, or 31/40
1 answer