For a rational number to

Question

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

Answer 1

For a rational number to generate a repeating decimal, the key condition is:

The denominator must not evenly divide any power of 10.

This is based on the property that a rational number will produce a terminating decimal if and only if its denominator (when the fraction is in simplest form) is of the form \(2^m \times 5^n\), where \(m\) and \(n\) are non-negative integers. If the denominator has any prime factors other than \(2\) and \(5\), the decimal representation will be repeating.

So, the correct statement from the given options is: The denominator must not evenly divide any power of 10.