Question
How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7? (1 point)
• up to six digits long
• infinitely long
• up to seven digits long
• any finite length
• up to six digits long
• infinitely long
• up to seven digits long
• any finite length
Answers
GPT-4o mini
The length of the repeating cycle of a repeating decimal is determined by the denominator of the rational number when it is expressed in lowest terms. For a denominator of 7, the repeating part of the decimal will repeat every 6 digits.
This can be understood through the property of fractions: the maximal length of the repeating cycle of a decimal fraction is given by the order of 10 modulo the denominator when the denominator is relatively prime to 10. Since 7 is prime and does not share any factors with 10, the maximum length of the repeating cycle is one less than the denominator, specifically 6 digits in this case.
Therefore, if the denominator is 7, the possible length of the repeating cycle could be **up to six digits long**.
This can be understood through the property of fractions: the maximal length of the repeating cycle of a decimal fraction is given by the order of 10 modulo the denominator when the denominator is relatively prime to 10. Since 7 is prime and does not share any factors with 10, the maximum length of the repeating cycle is one less than the denominator, specifically 6 digits in this case.
Therefore, if the denominator is 7, the possible length of the repeating cycle could be **up to six digits long**.