Look at several examples of rational numbers in the form (p/q) (q not equal to 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?

Look at several examples of rational numbers in the form $\frac{p}{q}(q \neq 0)$, 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?

Rational numbers in the form $\frac{p}{q}$, where $p$ and $q$ are integers with no common factors other than 1 and having terminating decimal expansions, require $q$ to have a specific property related to its prime factors.

Let's examine several examples of such rational numbers:

1. $\frac{1}{2}$ - Decimal expansion: 0.5
2. $\frac{7}{10}$ - Decimal expansion: 0.7
3. $\frac{3}{8}$ - Decimal expansion: 0.375
4. $\frac{25}{16}$ - Decimal expansion: 1.5625
5. $\frac{9}{40}$ - Decimal expansion: 0.225

From these examples, one can observe that for the decimal expansion to terminate, $q$ must be such that its prime factorization contains only 2, 5, or both. That is, any integer $q$ that leads to a terminating decimal must be of the form $2^m5^n$, where $m$ and $n$ are non-negative integers. This includes the case where $m$ or $n$ is zero, meaning $q$ could be a power of 2, a power of 5, or a product of powers of 2 and 5.

The reason behind this property is that when you simplify a fraction $\frac{p}{q}$ so that $p$ and $q$ have no common prime factors other than 1 and if $q$ has a terminating decimal expansion, it means that $q$ can only be made up of prime factors that are present in the base of the number system used for the decimal representation, which is 10 for our decimal system. Since 10 is equal to $2 \times 5$, $q$ must thus be divisible by only 2's, 5's, or both for its decimal expansion to terminate.