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Coprime Integers

Class 9 Mathematics के इस tag से जुड़े questions। हर question के साथ chapter, topic, level और difficulty दी गई है।

MathematicsIf \(\sqrt{2}\) is assumed to be rational and written as \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, why does a contradiction arise in the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsIn the proof by contradiction for the irrationality of \(\sqrt{2}\), what is the main purpose of assuming \(\sqrt{2}=\frac{p}{q}\) in lowest terms?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsSuppose \(\sqrt{3}=\frac{m}{n}\), where \(m\) and \(n\) are coprime integers. Which conclusion proves a contradiction to this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsWhich statement correctly describes the main idea used in the proof that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsIf \(\sqrt{3}\) is assumed to be \(\frac{p}{q}\) in lowest terms, where \(p\) and \(q\) are coprime integers, what follows from \(p^2=3q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsIn a proof by contradiction, Arjun assumes that \(\sqrt{2}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. He obtains \(m^2=2n^2\) and says, “\(m\) is even, so \(\sqrt{2}\) is irrational.” What is missing from his argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsIn a proof that √3 is irrational, suppose √3 = p/q, where p and q are coprime. Squaring gives p² = 3q², and on writing p = 3k, we get q² = 3k². Which conclusion is needed to complete the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. If \(p^2=3q^2\) is obtained, what is the correct next conclusion in the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, if assuming \(\sqrt{3}=\frac{p}{q}\) in lowest terms gives \(p^2=3q^2\), which conclusion creates the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsA student claims that if \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then \(p^2=3q^2\) proves only that \(p\) is divisible by 3. What is the correct improvement to the argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After obtaining \(p^2=2q^2\), which conclusion creates a contradiction in this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After obtaining \(p^2=3q^2\), which conclusion is correct?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsIf assuming \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime, leads to \(p^2=3q^2\), which conclusion creates the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsA student writes: If \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime, then \(a^2=3b^2\). The student concludes that 3 divides \(b\). What is the error in this conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsA student claims that \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Which correct conclusion follows from this claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. After concluding from \(p^2=3q^2\) that \(p\) is divisible by 3, which conclusion completes the contradiction proving \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsIn a proof by contradiction that \(\sqrt{3}\) is irrational, what fact produces the contradiction after assuming \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime integers?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsBefore assuming \(\sqrt{3}=\frac{p}{q}\) in a proof by contradiction that \(\sqrt{3}\) is irrational, which condition on \(p\) and \(q\) is essential?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsRiya assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring, she gets \(p^2=3q^2\). Which argument correctly proves a contradiction in this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsSuppose \\(\sqrt{3}=\frac{p}{q}\\), where p and q are coprime positive integers. In the proof that \\(\sqrt{3}\\) is irrational, which fact gives the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium