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)%0D%0AResponses%0D%0A%0D%0Aany finite length%0D%0Aany finite length%0D%0A%0D%0Aup to seven digits long%0D%0Aup to seven digits long%0D%0A%0D%0Aup to six digits long%0D%0Aup to six digits long%0D%0A%0D%0Ainfinitely long

Answer 1

The length of the repeating cycle of a decimal fraction is determined by the denominator when the fraction is in its simplest form. In the case of a denominator of \(7\), the fraction \( \frac{1}{7} \) gives rise to a repeating decimal.

To find the maximum length of the repeating cycle for \( \frac{1}{7} \), we can examine the decimal representation:

\[ \frac{1}{7} = 0.\overline{142857} \]

The repeating portion "142857" has a length of 6 digits. In general, the length of the repeating cycle can be up to \( n-1 \) where \( n \) is the number of distinct prime factors of the denominator, and since \( 7 \) is prime, we have \( n = 7 \).

Therefore, for a fraction with a denominator of \(7\), the repeating cycle can be up to 6 digits long.

Thus, the correct response is:

up to six digits long.