What Is A Counterexample Of The Conditional Statement "If A Number Is Divisible By Three, Then It Is Odd"? Options: A. 1, B. 3, C. 6, D. 9.

Jun 21, 2025 by ADMIN 140 views

Counterexample to the Conditional Statement If a Number Is Divisible by Three Then It Is Odd

In the realm of mathematics, conditional statements play a crucial role in forming logical arguments and theorems. A conditional statement, often expressed in the form "If P, then Q," asserts that if the condition P (the hypothesis) is true, then the conclusion Q must also be true. However, the validity of a conditional statement hinges on its resistance to counterexamples. A counterexample is a specific instance where the hypothesis P holds true, but the conclusion Q is false. Finding a counterexample effectively disproves the conditional statement. In this article, we will delve into the conditional statement "If a number is divisible by three, then it is odd" and explore potential counterexamples that challenge its validity. We will dissect the statement, analyze the properties of divisibility and odd numbers, and systematically evaluate given options to identify the counterexample. By understanding counterexamples, we gain a deeper appreciation for the nuances of conditional statements and their significance in mathematical reasoning.

To effectively identify a counterexample, it's crucial to grasp the essence of a conditional statement. A conditional statement follows the structure "If P, then Q," where P represents the hypothesis and Q represents the conclusion. The statement asserts that if P is true, then Q must also be true. The only scenario that invalidates a conditional statement is when P is true, but Q is false. This instance serves as a counterexample, demonstrating that the statement does not hold universally. Consider the statement "If it is raining, then the ground is wet." The hypothesis (P) is "it is raining," and the conclusion (Q) is "the ground is wet." A counterexample would be a situation where it is raining (P is true), but the ground is not wet (Q is false), such as rain falling on a covered surface. Identifying counterexamples is a fundamental skill in mathematical reasoning, enabling us to assess the validity of statements and refine our understanding of mathematical concepts. It encourages critical thinking and attention to detail, ensuring that our conclusions are well-supported by evidence and logic. Conditional statements are the backbone of mathematical proofs and logical arguments, and the ability to analyze them effectively is paramount in mathematical problem-solving.

Our focus is on the conditional statement: "If a number is divisible by three, then it is odd." Here, the hypothesis (P) is "a number is divisible by three," and the conclusion (Q) is "it is odd." To find a counterexample, we need to identify a number that satisfies the hypothesis (divisible by three) but does not satisfy the conclusion (not odd, i.e., even). To determine if a number is divisible by three, we check if it can be divided by three without leaving a remainder. Odd numbers are integers that cannot be divided evenly by two, while even numbers are integers that can be divided evenly by two. The statement implies that all numbers divisible by three are odd. However, this might not be universally true. Our goal is to find a number that can be divided by three and is also an even number. This would serve as a counterexample, proving that the conditional statement is false. The search for a counterexample requires careful consideration of both divisibility by three and the properties of odd and even numbers. By systematically analyzing different numbers, we can determine if the statement holds true or if a counterexample exists.

We are presented with four options: A. 1, B. 3, C. 6, and D. 9. To determine which option is a counterexample, we must test each one against the conditional statement "If a number is divisible by three, then it is odd." We need to identify a number that is divisible by three but is not odd (i.e., it is even).

Option A: 1

Is 1 divisible by three? No, 1 divided by 3 leaves a remainder. Therefore, 1 does not satisfy the hypothesis. Since the hypothesis is false, 1 cannot be a counterexample.
Option B: 3

Is 3 divisible by three? Yes, 3 divided by 3 equals 1 with no remainder. So, 3 satisfies the hypothesis. Is 3 odd? Yes, 3 is not divisible by 2. Therefore, 3 satisfies the conclusion. Since 3 satisfies both the hypothesis and the conclusion, it is not a counterexample.
Option C: 6

Divisibility by Three: Is 6 divisible by three? Yes, 6 divided by 3 equals 2 with no remainder. So, 6 satisfies the hypothesis. Even Number Check: Is 6 odd? No, 6 is divisible by 2 (6 divided by 2 equals 3). Therefore, 6 does not satisfy the conclusion. Since 6 satisfies the hypothesis (divisible by three) but does not satisfy the conclusion (odd), it is a counterexample.
Option D: 9

Is 9 divisible by three? Yes, 9 divided by 3 equals 3 with no remainder. So, 9 satisfies the hypothesis. Is 9 odd? Yes, 9 is not divisible by 2. Therefore, 9 satisfies the conclusion. Since 9 satisfies both the hypothesis and the conclusion, it is not a counterexample.

After analyzing each option, we found that the number 6 is a counterexample to the conditional statement "If a number is divisible by three, then it is odd." The number 6 is divisible by three, satisfying the hypothesis, but it is also an even number, failing to meet the conclusion that it must be odd. This single counterexample is sufficient to disprove the conditional statement, demonstrating that it does not hold true for all numbers divisible by three. The existence of the counterexample highlights the importance of careful consideration when making generalizations in mathematics. While a pattern may hold true in many cases, it only takes one counterexample to demonstrate that the statement is not universally valid. In this case, the number 6 serves as a clear and concise counterexample, solidifying our understanding of conditional statements and the nature of mathematical proof.

In conclusion, the counterexample to the conditional statement "If a number is divisible by three, then it is odd" is C. 6. This number effectively disproves the statement by demonstrating that a number can be divisible by three and yet still be even. This exercise underscores the significance of counterexamples in mathematical reasoning, as they serve as powerful tools to challenge and refine our understanding of mathematical statements. By identifying counterexamples, we strengthen our ability to construct valid arguments and avoid making generalizations that are not universally true. The process of analyzing conditional statements and searching for counterexamples encourages critical thinking and attention to detail, essential skills for anyone engaging with mathematics. Furthermore, this exploration highlights the importance of precise language and definitions in mathematics. The terms "divisible by three," "odd," and "conditional statement" each have specific meanings, and understanding these meanings is crucial for evaluating the validity of mathematical claims. The ability to identify counterexamples is a valuable asset in mathematical problem-solving and serves as a reminder that mathematical truths must be rigorously tested and verified. Therefore, the correct answer is C. 6, which serves as a clear and concise demonstration that the conditional statement is not universally true.