In the proof of \(\sqrt{2}\), if both \(m\) and \(n\) are even, what can be said about their highest common factor?
Answer and explanation
Correct answer: The highest common factor will be at least \(2\)
If both \(m\) and \(n\) are even, then we can write \(m=2p\) and \(n=2q\) for some integers \(p,q\). Thus, \(2\) is a common factor of both \(m\) and \(n\), so their highest common factor is at least \(2\). In the irrationality proof of \(\sqrt{2}\), this contradicts the assumption that \(m\) and \(n\) are coprime. Exam tip: if two integers are both even, their HCF cannot be \(1\).
Frequently asked questions
What is the correct answer to this question?
The highest common factor will be at least \(2\)
Why is this the correct answer?
If both \(m\) and \(n\) are even, then we can write \(m=2p\) and \(n=2q\) for some integers \(p,q\). Thus, \(2\) is a common factor of both \(m\) and \(n\), so their highest common factor is at least \(2\). In the irrationality proof of \(\sqrt{2}\), this contradicts the assumption that \(m\) and \(n\) are coprime. Exam tip: if two integers are both even, their HCF cannot be \(1\).
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.
