In the proof of √3, 3 | h² implies 3 | h. Which broader principle does this illustrate?
Answer and explanation
Correct answer: A prime factor occurs in a square only if it occurs in the original number
The governing principle is unique prime factorisation. If the prime 3 divides h², then the prime factor 3 must already occur in the factorisation of h; squaring only doubles its exponent and cannot create a new prime factor. Therefore 3 divides h. The other options are false general statements and do not support the irrationality proof.
Frequently asked questions
What is the correct answer to this question?
A prime factor occurs in a square only if it occurs in the original number
Why is this the correct answer?
The governing principle is unique prime factorisation. If the prime 3 divides h², then the prime factor 3 must already occur in the factorisation of h; squaring only doubles its exponent and cannot create a new prime factor. Therefore 3 divides h. The other options are false general statements and do not support the irrationality proof.
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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