Rational Numbers in Different Forms Quick Check

Question

2 of 52 of 5 Items
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

up to seven digits long
up to seven digits long

any finite length
any finite length

infinitely long
infinitely long

up to six digits long
up to six digits long
Skip to navigation

Answer 1

The repeating cycle of a repeating decimal for a rational number is determined by the denominator when it is in its simplest form. For a denominator of 7, the cycle can be up to 6 digits long.

This is based on the property that the maximum length of the repeating cycle of a decimal representation of a rational number \( \frac{p}{q} \) occurs at \( q - 1 \) if \( q \) is prime and does not divide 10. Since 7 is a prime number, the repeating cycle can be up to 6 digits long.

So the correct answer is: up to six digits long.