For a rational number to generate a repeating decimal what must be true

Answer 1

For a rational number to generate a repeating decimal, its denominator must contain prime factors other than 2 and/or 5 when the rational number is expressed in its simplest form.

Here’s the reasoning:

1. A rational number can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\).
2. If the denominator \(q\) (in its simplest form) has only the prime factors 2 and/or 5, then the decimal representation of the fraction will terminate. For example, \(\frac{1}{2} = 0.5\) and \(\frac{1}{8} = 0.125\).
3. However, if the denominator contains prime factors other than 2 and 5, then the decimal representation will be repeating. For example, \(\frac{1}{3} = 0.333...\) and \(\frac{1}{6} = 0.1666...\).

In summary, a rational number generates a repeating decimal if, when expressed in simplest form, the denominator has at least one prime factor other than 2 or 5.