A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of \(q\), when this number is expressed in the form \(\dfrac{p}{q}\)? Give reasons.
Since 327.7081 is a terminating decimal number, \(q\) must be of the form \(2^m 5^n\), where \(m,n\) are natural numbers (non-negative integers).
Step 1: Recall the property of terminating decimals.
A rational number \(\dfrac{p}{q}\) in lowest terms has a terminating decimal expansion if and only if the denominator \(q\) has no prime factors other than 2 or 5.
Step 2: Connect with powers of 10.
Any terminating decimal can be written as a fraction with denominator \(10^k\) for some integer \(k\).
For example, \(327.7081 = \dfrac{3277081}{10000}\).
Since \(10000 = 10^4 = 2^4 \times 5^4\), the denominator has only 2 and 5 as prime factors.
Step 3: Simplify and check denominator form.
Even if we reduce \(\dfrac{3277081}{10000}\) to lowest terms, the denominator will still be of the form \(2^m 5^n\).
This is because cancelling common factors between numerator and denominator cannot introduce new primes in the denominator.
Conclusion.
The prime factors of \(q\) are only 2 and 5, i.e., \(q = 2^m 5^n\) for some non-negative integers \(m,n\).