A fraction in the form of \( \frac{a}{b} \) (where \( a \) is a whole number greater than 0 and \( b \) is the denominator) will always convert to a repeating decimal if the denominator \( b \) has any prime factors other than 2 or 5.
In other words, for a fraction to convert to a terminating decimal, its denominator must only consist of the prime factors 2 and/or 5. If it includes any other prime factor (such as 3, 7, 11, etc.), it will convert to a repeating decimal.
For example:
- The fraction \( \frac{1}{2} \) and \( \frac{1}{5} \) are terminating decimals (0.5 and 0.2, respectively).
- The fraction \( \frac{1}{3} \) or \( \frac{1}{7} \) results in repeating decimals (0.333... and 0.142857..., respectively).
So, to ensure that the fraction always converts to a repeating decimal, the denominator should have at least one prime factor other than 2 or 5. For example, \( b = 3, 6, 7, 9, 11, \) etc. will lead to a repeating decimal.