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

up to six digits long
up to six digits long

up to seven digits long
up to seven digits long

any finite length

Answer 1

The repeating cycle of a repeating decimal for a rational number depends on the denominator after it has been simplified. For a denominator of 7, the length of the repeating cycle can be determined by finding the order of 10 modulo 7.

When you divide by 7, you can see that 10^1 (which is 10) mod 7 gives a remainder of 3, 10^2 gives a remainder of 2, 10^3 gives a remainder of 6, 10^4 gives a remainder of 4, and 10^5 gives a remainder of 5, while 10^6 gives a remainder of 1. This means that the order is 6, suggesting that the repeating decimal can have a cycle of up to 6 digits long.

Thus, the answer is:
**up to six digits long**.