Question

For a rational number to generate a repeating decimal, what must be true?
A. The rational number must not be a mixed number
B. The denominator must be a prime number
C. The denominator must not evenly divide any power of 10
D. The numerator must be an odd number

Answer 1

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.**