Question

if a repeating decimal has a repeating cycle of three digits it will convert to a rational number with what denominator

Answer 1

A repeating decimal with a repeating cycle of three digits can be converted to a rational number with a denominator that is a power of 10 minus 1, specifically \(999\) for a three-digit repeating cycle.

To elaborate, if you have a repeating decimal like \(0.\overline{xyz}\) (where \(xyz\) are the three repeating digits), you can express it as:

\[
\frac{xyz}{999}
\]

For example, for the decimal \(0.\overline{123} = 0.123123123...\), it can be converted to the fraction \(\frac{123}{999}\).

Thus, the denominator for a repeating decimal with a three-digit repeating cycle is always \(999\).