Which option gives the correct reasoning from (p²) to (p) in the proof of (√3)?
Answer and explanation
Correct answer: If (p²) is divisible by (3), then (p) is divisible by (3)
The governing fact is Euclid’s lemma for a prime: if a prime divides the square of an integer, it divides that integer itself. In the √3 proof, the equation p² = 3q² shows that 3 divides p². Because 3 is prime, it follows that 3 divides p, so p can be written as p = 3k. This is the required step and makes option B correct. The conclusion does not say that p is divisible by 2, equal to zero, or equal to q. After this step, substitution is used to show that q is also divisible by 3, producing the contradiction with lowest terms.
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What is the correct answer to this question?
If (p²) is divisible by (3), then (p) is divisible by (3)
Why is this the correct answer?
The governing fact is Euclid’s lemma for a prime: if a prime divides the square of an integer, it divides that integer itself. In the √3 proof, the equation p² = 3q² shows that 3 divides p². Because 3 is prime, it follows that 3 divides p, so p can be written as p = 3k. This is the required step and makes option B correct. The conclusion does not say that p is divisible by 2, equal to zero, or equal to q. After this step, substitution is used to show that q is also divisible by 3, producing the contradiction with lowest terms.
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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