Why is writing (p=3q) from (p^2=3q^2) unacceptable in the proof of (\sqrt{3})?
Answer and explanation
Correct answer: Because the correct conclusion is (3\mid p), not (p=3q)
From p^2=3q^2, the right conclusion is that 3 divides p. This follows from prime-factor reasoning: the exponent of the prime 3 in a perfect square is even, while the factor 3 on the right makes the exponent in 3q^2 odd unless q also supplies a factor of 3. Thus p must contain a factor 3. However, this does not mean p=3q.
The symbols p and q represent the numerator and denominator in the assumed fraction, and q need not equal p divided by 3. The proof normally writes p=3k for some integer k, then substitutes this into the equation to show that 3 also divides q, producing the contradiction. Therefore option A is correct. The other choices do not describe the valid divisibility argument.
Frequently asked questions
What is the correct answer to this question?
Because the correct conclusion is (3\mid p), not (p=3q)
Why is this the correct answer?
From p^2=3q^2, the right conclusion is that 3 divides p. This follows from prime-factor reasoning: the exponent of the prime 3 in a perfect square is even, while the factor 3 on the right makes the exponent in 3q^2 odd unless q also supplies a factor of 3. Thus p must contain a factor 3. However, this does not mean p=3q.
The symbols p and q represent the numerator and denominator in the assumed fraction, and q need not equal p divided by 3. The proof normally writes p=3k for some integer k, then substitutes this into the equation to show that 3 also divides q, producing the contradiction. Therefore option A is correct. The other choices do not describe the valid divisibility argument.
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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