Which of the following rational numbers can be expressed as a repeating decimal: 7/8, 5/12, 17/25, or 31/40? (1 point)
To determine whether a rational number can be expressed as a repeating decimal, we need to check the denominator for factors other than 2 and 5.
The prime factorization of 8 is $2^3$, which means 8 has factors of 2, but no factors of 5. Therefore, 7/8 cannot be expressed as a repeating decimal.
The prime factorization of 12 is $2^2\cdot3$, which means 12 has factors of 2 and 3, but no factors of 5. Therefore, 5/12 cannot be expressed as a repeating decimal.
The prime factorization of 25 is $5^2$, which means 25 has factors of 5, but no factors of 2. Therefore, 17/25 can be expressed as a repeating decimal.
The prime factorization of 40 is $2^3\cdot5$, which means 40 has factors of 2 and 5. Therefore, 31/40 cannot be expressed as a repeating decimal.
Therefore, the rational numbers that can be expressed as a repeating decimal are 7/8 and 31/40
To determine which of the given rational numbers can be expressed as a repeating decimal, we need to consider the factors of the denominators.
Let's analyze each option:
1. For 7/8, the denominator is 8, which is not divisible by any prime number other than 2. Therefore, it can be expressed as a terminating decimal.
2. For 5/12, the denominator is 12. It can be factored into 2 * 2 * 3. Since 2 and 5 are relatively prime, this fraction can be expressed as a repeating decimal.
3. For 17/25, the denominator is 25, which is divisible by 5 squared. Since 17 and 5 are relatively prime, this fraction can be expressed as a repeating decimal.
4. For 31/40, the denominator is 40, which can be factored into 2^3 * 5. Since 31 and 2 * 5 = 10 are relatively prime, this fraction can be expressed as a repeating decimal.
Therefore, the rational numbers 5/12, 17/25, and 31/40 can be expressed as repeating decimals.