How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?(1 point)

Question

Responses

infinitely long
infinitely long

up to seven digits long
up to seven digits long

up to six digits long
up to six digits long

any finite length

Answer 1

The repeating cycle of a repeating decimal for a rational number depends on the denominator when the fraction is in its simplest form. For a fraction with a denominator of 7, we can determine the maximum possible length of the repeating cycle by examining the properties of the number.

Since 7 is a prime number and does not divide any power of 10, the length of the repeating cycle can be calculated using the formula which states that the maximum length of the repeating decimal cycle is equal to the order of 10 modulo 7.

Calculating it results in the order of 10 modulo 7 being 6, because \( 10^1 \mod 7 = 3 \), \( 10^2 \mod 7 = 2 \), \( 10^3 \mod 7 = 6 \), \( 10^4 \mod 7 = 4 \), \( 10^5 \mod 7 = 5 \), and \( 10^6 \mod 7 = 1 \), repeating thereafter.

Thus, the length of the repeating cycle could be up to 6 digits long.

The correct response is: up to six digits long.