Which of the following rational numbers can be expressed as a repeating decimal: 78 , 512 , 1725 , or 3140 ?(1 point) $$
To determine if a rational number can be expressed as a repeating decimal, we need to check if its denominator has any prime factors other than 2 or 5.
The prime factorization of 78 is $2\cdot3\cdot13$. Since the denominator contains the prime factor 3, 78 cannot be expressed as a repeating decimal.
The prime factorization of 512 is $2^9$. Since the denominator only contains the prime factor 2, 512 can be expressed as a repeating decimal.
The prime factorization of 1725 is $3^3 \cdot 5\cdot 5\cdot 23$. Since the denominator contains the prime factor 3, 1725 cannot be expressed as a repeating decimal.
The prime factorization of 3140 is $2^2\cdot5\cdot157$. Since the denominator only contains the prime factors 2 and 5, 3140 can be expressed as a repeating decimal.
Thus, the rational numbers that can be expressed as repeating decimals are 512 and 3140.