Decide whether the statement Every rational number is an integer is correct or incorrect, based on the formal definitions of rational numbers and integers.

Introduction / Context:
This conceptual question tests knowledge of the definitions and relationships between rational numbers and integers. Rational numbers are a broad class of numbers that include fractions and decimals, while integers are whole numbers that can be positive, negative, or zero. The statement Every rational number is an integer must be evaluated carefully, because it suggests that the set of rational numbers is contained within the set of integers. Understanding the actual hierarchy of number sets is crucial in number theory and algebra.

Given Data / Assumptions:
- A rational number is defined as any number that can be written in the form p / q where p and q are integers and q is not zero.
- Integers include all whole numbers and their negatives, such as ..., -3, -2, -1, 0, 1, 2, 3, and so on.
- The given statement claims that every rational number must be an integer.
- We assume standard definitions from school level mathematics.

Concept / Approach:
To test a universal statement such as Every rational number is an integer, it is enough to find a single counterexample: a rational number that is not an integer. If such a number exists, the entire statement is incorrect. Many fractions like 1/2, 3/4, or 5/6 are rational because they can be expressed as ratios of integers, yet they are not integers. Therefore, checking any one such fraction is sufficient to disprove the statement.

Step-by-Step Solution:
Step 1: Recall the definition of rational number: any number that can be written as p / q where p and q are integers and q is not zero. Step 2: Recall the definition of integer: a number with no fractional or decimal part, such as -2, -1, 0, 1, 2, 3. Step 3: Observe that every integer n can indeed be written as n / 1, so every integer is a rational number. Step 4: However, consider a number like 1/2. It can be written as 1 / 2 where 1 and 2 are integers and 2 is not zero, so 1/2 is a rational number. Step 5: At the same time, 1/2 is not an integer because it has a fractional part and lies strictly between 0 and 1. Step 6: This single counterexample shows that there exists at least one rational number that is not an integer. Step 7: Therefore the statement Every rational number is an integer is incorrect.

Verification / Alternative check:
We can produce many other rational numbers that are not integers, such as 3/4, -5/2, or 7/10. They all fit the rational definition but clearly are not whole numbers. The proper relationship is that the set of integers is a subset of the set of rational numbers, not the other way around. In symbols, every integer is rational, but not every rational number is an integer. This confirms that the statement in the question reverses the correct inclusion and is false.

Why Other Options Are Wrong:
Option Correct statement: This would be true only if every rational number were an integer. The counterexample 1/2 disproves this, so this option is wrong.
Option The statement cannot be determined: The definitions are precise and standard; there is no ambiguity, so the truth value can be determined exactly.
Option The statement depends on the numerator and denominator: Although some rational numbers with denominator 1 are integers, the existence of many others with denominator greater than 1 shows that the statement is not conditionally true but simply false as written.
Option True only for positive rational numbers: Positive rational numbers like 1/2 and 3/4 are not integers, so restricting to positive numbers does not fix the problem.

Common Pitfalls:
A frequent misunderstanding is to mix up the direction of inclusion, thinking that rational numbers are a subset of integers instead of the opposite. Some learners also assume that rational means reasonable or simple and therefore always a whole number. It is important to remember that rational specifically refers to ratios of integers. Visualizing the number line and placing fractions between integers helps make the distinction clearer.

Final Answer:
The statement Every rational number is an integer is an incorrect statement.