For a rational number to generate a repeating decimal, we need to consider the properties of its denominator when expressed in simplest form. A rational number \( \frac{a}{b} \) (where \( a \) is the numerator and \( b \) is the denominator) will have a repeating decimal if the denominator \( b \) contains prime factors other than 2 and 5.
This is because a decimal will terminate if the denominator (after simplifying the fraction) can be expressed as \( 2^m \times 5^n \) for non-negative integers \( m \) and \( n \). If there are other prime factors, the decimal will repeat.
Given this information, let's analyze the options provided:
A. The rational number must not be a mixed number - This is not necessarily true; mixed numbers can also produce repeating decimals.
B. The denominator must be a prime number - This is not true because prime denominators like 3, 7, etc. can produce repeating decimals, but not all repeating decimals must be caused by prime denominators.
C. The denominator must not evenly divide any power of 10 - This is correct. If the denominator has factors other than 2 or 5, it will not divide any power of 10 and hence will produce a repeating decimal.
D. The numerator must be an odd number - This is not relevant to whether the decimal representation is repeating or terminating.
Thus, the correct answer is: C. The denominator must not evenly divide any power of 10.
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