To solve this problem, let's analyze the sum of two rational numbers and the properties that might apply.
Consider two rational numbers:
[tex]\[
\frac{a}{b} \quad \text{and} \quad \frac{c}{d}
\][/tex]
where [tex]\( a, b, c, \)[/tex] and [tex]\( d \)[/tex] are integers and [tex]\( b \)[/tex] and [tex]\( d \)[/tex] are non-zero.
The sum of these two rational numbers is:
[tex]\[
\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
\][/tex]
Here, [tex]\(ad + bc\)[/tex] and [tex]\(bd\)[/tex] are both integers because they are the result of multiplying and adding integers, which always yields integers.
Given the properties provided:
1. Associative Property: This property states that the way in which numbers are grouped when adding or multiplying does not change their sum or product. This property does not specifically address whether [tex]\(ad + bc\)[/tex] and [tex]\(bd\)[/tex] are integers.
2. Closure Property: This property states that performing an operation (such as addition or multiplication) on any two numbers within a set (such as the set of integers) will produce another number within that set. In this context, it asserts that the sum and product of any two rational numbers are also rational numbers because their numerator and denominator remain integers.
3. Commutative Property: This property states that the order in which you add or multiply two numbers does not change the result, i.e., [tex]\(a + b = b + a\)[/tex] or [tex]\(ab = ba\)[/tex]. This also does not address the specific integers [tex]\(ad + bc\)[/tex] and [tex]\(bd\)[/tex].
Given our situation, the pertinent property is that the numbers [tex]\(ad + bc\)[/tex] and [tex]\(bd\)[/tex] are integers because they are operations within the set of integers (specifically, closure under addition and multiplication).
Therefore, the correct option that identifies this property is:
Option 2: Closure Property
Hence, the answer to the question is:
[tex]\[
\boxed{2}
\][/tex]
