In the proof of √3, which statement is a middle step rather than the final conclusion?
Answer and explanation
Correct answer: p is divisible by 3
The proof assumes, for contradiction, that √3 = p/q in lowest terms. From p² = 3q², one first concludes that 3 divides p, so p = 3k. This is only an intermediate inference, represented by option C. Substituting p = 3k then gives 9k² = 3q² and hence q² = 3k², which shows that 3 also divides q. Since both p and q are divisible by 3, the fraction was not in lowest terms. That contradiction rejects the original rational assumption, and the final conclusion is that √3 is irrational. Options A, B, and D describe the concluding part rather than the requested middle step.
Frequently asked questions
What is the correct answer to this question?
p is divisible by 3
Why is this the correct answer?
The proof assumes, for contradiction, that √3 = p/q in lowest terms. From p² = 3q², one first concludes that 3 divides p, so p = 3k. This is only an intermediate inference, represented by option C. Substituting p = 3k then gives 9k² = 3q² and hence q² = 3k², which shows that 3 also divides q. Since both p and q are divisible by 3, the fraction was not in lowest terms. That contradiction rejects the original rational assumption, and the final conclusion is that √3 is irrational. Options A, B, and D describe the concluding part rather than the requested middle step.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Proof of irrationality of square root 2 and square root 3.
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