If p is not divisible by 3 in p² = 3q², what contradiction appears?
Answer and explanation
Correct answer: p² should not be divisible by 3 but the equation makes it divisible
The correct answer is A. If p is not divisible by 3, then its prime factorisation contains no factor 3. Squaring p does not introduce a new prime factor, so p² is also not divisible by 3. However, the equation p² = 3q² writes the left side as three times an integer square, so the right-hand expression is divisible by 3; consequently p² must be divisible by 3. This is a direct contradiction. In the standard irrationality proof, one first assumes √3 = p/q in lowest terms, with q nonzero and p, q coprime. The contradiction forces both numbers to be divisible by 3, violating the lowest-form assumption. The other options do not follow from the equation.
Frequently asked questions
What is the correct answer to this question?
p² should not be divisible by 3 but the equation makes it divisible
Why is this the correct answer?
The correct answer is A. If p is not divisible by 3, then its prime factorisation contains no factor 3. Squaring p does not introduce a new prime factor, so p² is also not divisible by 3. However, the equation p² = 3q² writes the left side as three times an integer square, so the right-hand expression is divisible by 3; consequently p² must be divisible by 3. This is a direct contradiction. In the standard irrationality proof, one first assumes √3 = p/q in lowest terms, with q nonzero and p, q coprime. The contradiction forces both numbers to be divisible by 3, violating the lowest-form assumption. The other options do not follow from the equation.
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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