Properties — closure, commutative, associative, distributive

A set is closed under an operation if performing that operation on any two members always produces a result that is also in the set. Here we are asking about integers under multiplication.

For example, $3\times (-4)=-12$, which is an integer. Also, $(-7)\times 5=-35$, also an integer. No matter which two integers we multiply, the result is always an integer.

The closure property under multiplication means: multiplying two integers always gives an integer. This matches the first option exactly.

We need the statement that correctly describes closure of integers under multiplication — meaning the result stays in the set of integers.

'Multiplying two integers always gives a positive number' is false. For example, $3\times (-2)=-6$, which is negative. Eliminate this option.

'The order in which you multiply integers does not matter' describes the commutative property, not closure. Eliminate this option.

'Multiplying an integer by 1 leaves it unchanged' describes the multiplicative identity, not closure. Eliminate this option.

The remaining option — 'Multiplying two integers always gives an integer' — correctly describes closure: the result stays within the set of integers.

For closure to hold, the operation must always produce a result in the set — not just sometimes. One counterexample is enough to disprove closure.

Consider $3÷4=0.75$. Both 3 and 4 are integers, but $0.75$ is not an integer. This single counterexample proves integers are not closed under division.

The student used one case where division happened to give an integer ($6÷2=3$), but closure requires it to work for all pairs of integers, not just some. The student's reasoning is flawed.

The correct response points out that $3÷4=0.75$ is a counterexample, showing the claim is incorrect.

We need to decide whether the student's claim (integers are closed under division) is valid, and find the best evaluation.

'The claim is correct because dividing two integers sometimes gives an integer' and 'The claim is correct because 3 is an integer' both agree with the claim. But closure requires the result to always be an integer, not just sometimes. Eliminate both.

'The claim is incorrect because division is not commutative' confuses two different properties — commutativity and closure. Lack of commutativity is irrelevant to closure. Eliminate this option.

'The claim is incorrect because $3÷4=0.75$, which is not an integer' is the only response that correctly uses a counterexample to disprove the closure claim.