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 seven digits long

any finite length

Orifinitely long

up to six digies long

Answer 1

For a rational number, the length of the repeating cycle of its decimal representation is determined by the denominator (after simplifying the fraction). When the denominator is 7, the repeating decimal can have a maximum length of up to 6 digits.

This is because the possible lengths of the repeating part of a decimal can be determined by the factors of 10 (the base of our number system) in relation to the denominator. More specifically, a decimal will repeat after at most \( n \) digits, where \( n \) is the smallest integer such that \( 10^n \equiv 1 \mod d \), with \( d \) being the denominator after removing powers of 2 and 5.

For 7, the length of the repeating part of the decimal is 6.

So the correct answer is:

**up to six digits long**.