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) Responses infinitely long infinitely long any finite length any finite length up to six digits long up to six digits long up to seven digits long
Answers
GPT-4o mini
The repeating cycle of a decimal that results from dividing by a rational number can be determined by the factors of the denominator. For a fraction \( \frac{1}{7} \), the decimal representation is \( 0.142857\ldots \), which has a repeating cycle of 6 digits (142857).
In general, the maximum length of the repeating cycle of a decimal for a denominator \( d \) can be equal to \( d - 1 \) when \( d \) is a prime number. Since 7 is a prime number, the maximum length of the repeating cycle can be at most 6 digits long.
Thus, the correct response is: **up to six digits long**.
In general, the maximum length of the repeating cycle of a decimal for a denominator \( d \) can be equal to \( d - 1 \) when \( d \) is a prime number. Since 7 is a prime number, the maximum length of the repeating cycle can be at most 6 digits long.
Thus, the correct response is: **up to six digits long**.