For a rational number (which is a fraction expressed in the form \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator) to generate a repeating decimal, a key condition involving the denominator must be met.
The decimal representation of a rational number will be repeating if and only if the denominator (when the fraction is in lowest terms) contains prime factors other than \(2\) and \(5\). This is because the decimal representation of a fraction can be fully periodic (repeating) if the only primes in the denominator are those that divide powers of \(10\) (which is \(2^m \times 5^n\) for some non-negative integers \(m\) and \(n\)).
Therefore, the correct condition to determine if a fraction will generate a repeating decimal is:
- The denominator must not be expressible solely as a product of \(2\) and/or \(5\).
Among the options given in your question, the closest one is:
- The denominator must not even divide any power of (10.
This captures the idea that if the denominator contains any other prime besides \(2\) or \(5\), the decimal will be repeating.
So, to directly answer your inquiry: the requirement is tied to the prime factors of the denominator, specifically that it should not be limited to just \(2\) and \(5\).
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