What is the idea of prime factors of a perfect square in the proof of √2?
Answer and explanation
Correct answer: In a perfect square, the exponent of 2 must be even
The governing concept is the prime-factorization property of a perfect square: every prime occurs with an even exponent. In the contradiction proof, suppose √2 = x/y, where x and y are coprime integers and y is non-zero. Squaring gives x² = 2y². Therefore x² has an odd contribution from the factor 2 on the right, so 2 divides x; write x = 2k. Substitution gives 4k² = 2y², hence y² = 2k², so 2 also divides y. This contradicts the assumption that x and y have no common factor. Thus option A states the essential idea; the other choices make false universal claims or give the incorrect value of √2.
Frequently asked questions
What is the correct answer to this question?
In a perfect square, the exponent of 2 must be even
Why is this the correct answer?
The governing concept is the prime-factorization property of a perfect square: every prime occurs with an even exponent. In the contradiction proof, suppose √2 = x/y, where x and y are coprime integers and y is non-zero. Squaring gives x² = 2y². Therefore x² has an odd contribution from the factor 2 on the right, so 2 divides x; write x = 2k. Substitution gives 4k² = 2y², hence y² = 2k², so 2 also divides y. This contradicts the assumption that x and y have no common factor. Thus option A states the essential idea; the other choices make false universal claims or give the incorrect value of √2.
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.
Student feedback
Was this question useful?
👍 0 Helpful 👎 0 Not helpful
Yes 0% No 0%
0 responsesStudent Reviews
No published reviews yet.
