If x and y are real numbers and the quotient x / y is an integer, which of the following statements need NOT always be true?

Introduction / Context:
This logical reasoning question explores what must or must not be true when a quotient x / y is known to be an integer. It is important to distinguish between conditions that are necessary for the quotient to be an integer and additional conditions that may or may not hold, such as the sign of the numbers.

Given Data / Assumptions:
The quotient x / y is an integer.x and y are real numbers, with y not equal to zero.We must identify which statement is not required to be true in every such situation.

Concept / Approach:
If x / y is an integer, then x can be written as y times some integer k, that is, x = k y. From this, we can infer that x is divisible by y and that y cannot be zero. However, nothing in this relationship forces x and y to be positive, because the integer k can be positive or negative and y can be positive or negative as long as their product produces x. We examine each statement in this light.

Step-by-Step Solution:
Step 1: From x / y being an integer, write x = k y, where k is an integer and y is nonzero.Step 2: This directly implies that x is divisible by y, so statement A must always be true.Step 3: Because division by zero is undefined, y must be a nonzero number for x / y to even exist, so statement B must also always be true.Step 4: Statement C simply repeats the given information that x / y is an integer, so it is trivially true whenever the condition is satisfied.Step 5: Statement D claims that both x and y are positive. This is not necessary. For example, if x = −6 and y = −3, then x / y = 2, which is still an integer, even though neither x nor y is positive.

Verification / Alternative check:
Test another example for statement D. Take x = 10 and y = −5. Then x / y = −2, which is also an integer. Here x is positive and y is negative, again contradicting the claim that both must be positive. These counterexamples show that statement D need not always hold, even though the quotient is an integer.

Why Other Options Are Wrong:
Statement A must hold because if x / y = k (integer), then x = k y, so x is exactly divisible by y. Statement B is necessary because dividing by zero is impossible. Statement C restates the given condition and is therefore always true under the problem assumption. They cannot be the answers to a need not always true question.

Common Pitfalls:
Some learners assume that integer division requires positive numbers, which is not correct. Integers include negative values and zero, and the quotient of two negative numbers can be a positive integer. Confusing optional conditions like positivity with necessary conditions like nonzero divisor often leads to incorrect choices.

Final Answer:
The statement that need not always be true is that both x and y are positive.