Look at several examples of rational numbers in the form `(p)/(q) (q ne 0)` where p and q are integers with no common factors other than `1` and having terminating decimal representations (expansions ). Can you guess what property q must satisfy ?

Answer

To determine the property that \( q \) must satisfy for the fraction \( \frac{p}{q} \) to have a terminating decimal representation, we can follow these steps:

### Step 1: Understand Terminating Decimals
A decimal representation is said to be terminating if it ends after a certain number of digits. For example, \( 0.5 \) and \( 0.75 \) are terminating decimals.

### Step 2: Identify the Form of Rational Numbers
We are given that \( p \) and \( q \) are integers with no common factors other than 1. This means that \( \frac{p}{q} \) is in its simplest form.

### Step 3: Analyze the Denominator \( q \)
For a fraction \( \frac{p}{q} \) to have a terminating decimal representation, the denominator \( q \) (when in its simplest form) must only have the prime factors 2 and/or 5. This is because the decimal system is base 10, and \( 10 = 2 \times 5 \).

### Step 4: Examples
Let’s consider some examples:
- \( \frac{1}{2} \) has a terminating decimal \( 0.5 \) (since \( q = 2 \)).
- \( \frac{3}{5} \) has a terminating decimal \( 0.6 \) (since \( q = 5 \)).
- \( \frac{1}{4} \) has a terminating decimal \( 0.25 \) (since \( q = 4 = 2^2 \)).
- \( \frac{1}{10} \) has a terminating decimal \( 0.1 \) (since \( q = 10 = 2 \times 5 \)).
- \( \frac{1}{3} \) does not have a terminating decimal (since \( q = 3 \) which is not a product of 2s and 5s).

### Conclusion
Thus, we conclude that for \( \frac{p}{q} \) to have a terminating decimal representation, \( q \) must be of the form \( 2^m \times 5^n \) where \( m \) and \( n \) are non-negative integers.