Q6:

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?

Step 1: Theoretical Foundation & Initial Setup

We are tasked with analyzing rational numbers of the form $\frac{p}{q}$, where:

• $p$ and $q$ are integers.
• $q \neq 0$.
• $\gcd(p, q) = 1$ [This indicates that $p$ and $q$ are coprime; they share no common factors other than 1, meaning the fraction is in its simplest form].
• The decimal representation of $\frac{p}{q}$ is terminating (it has a finite number of digits after the decimal point).

Our objective is to deduce the specific mathematical property that the denominator $q$ must satisfy to guarantee a terminating decimal expansion.

Step 2: Empirical Observation & Data Collection

To identify the pattern, we will generate several examples of rational numbers that yield terminating decimals, ensure they are in their simplest form, and examine the prime factorization of their denominators.

Step 3: Analyzing the Denominators ($q$)

By observing the "Prime Factorization of $q$" column, a distinct pattern emerges. The prime factors of every denominator $q$ in these terminating examples consist exclusively of the numbers $2$ and $5$.

To verify this exclusivity, let us briefly look at fractions where the denominator contains prime factors other than $2$ or $5$:

• $\frac{1}{3} = 0.3333...$ [Non-terminating, repeating; prime factor is $3$]
• $\frac{1}{7} = 0.142857...$ [Non-terminating, repeating; prime factor is $7$]
• $\frac{5}{6} = \frac{5}{2 \times 3} = 0.8333...$ [Non-terminating, repeating; prime factors are $2$ and $3$]

Step 4: Theoretical Justification

Any terminating decimal can be expressed as a fraction where the denominator is a power of $10$ (i.e., $10^k$ for some integer $k \geq 1$).

[Per the Fundamental Theorem of Arithmetic], the prime factorization of $10$ is strictly $2 \times 5$. Therefore, the prime factorization of any power of $10$ is:

$10^k = (2 \times 5)^k = 2^k \times 5^k$

When a fraction $\frac{x}{10^k}$ is reduced to its simplest form $\frac{p}{q}$ by canceling common factors, the only prime factors that can possibly remain in the denominator $q$ are $2$ and $5$. No new prime factors can be introduced during simplification.

Step 5: Visualizing the Logical Flow

The following diagram illustrates the structural transformation from a terminating decimal to its simplest fractional form, proving why $q$ is restricted to powers of $2$ and $5$.

Step 6: Formulating the General Property

Based on the empirical data and theoretical proof, for any rational number $\frac{p}{q}$ (where $p$ and $q$ are coprime) to have a terminating decimal expansion, the prime factorization of the denominator $q$ must not contain any prime numbers other than $2$ and $5$.

Mathematically, this is expressed as:

$q = 2^m \times 5^n$

where $m$ and $n$ are non-negative integers ($m, n \geq 0$).

Final Solution: For a rational number $\frac{p}{q}$ (where $p$ and $q$ are coprime integers) to have a terminating decimal representation, the prime factorization of the denominator $q$ must consist exclusively of powers of $2$ and/or $5$. That is, $q$ must satisfy the property $q = 2^m \times 5^n$, where $m$ and $n$ are non-negative integers.