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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।

MathematicsIn the proof by contradiction that \(\sqrt{3}\) is irrational, if \(p\) and \(q\) are coprime and \(p^2=3q^2\), which conclusion follows immediately?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, assume that \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. Which conclusion creates a contradiction to the initial assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhich statement is correct in the proof of the irrationality of \(\sqrt{3}\), based on \(a^2=3b^2\), where \(a\) and \(b\) are coprime integers?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsIn the proof by contradiction, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Why does this assumption lead to a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsSuppose \\(\sqrt{3}=\frac{p}{q}\\), where \\(p\\) and \\(q\\) are coprime positive integers. Which conclusion from \\(p^2=3q^2\\) gives the contradiction needed to prove that \\(\sqrt{3}\\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, if it is shown that 3 divides both \(a\) and \(b\), which conclusion creates the contradiction in the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsA student assumes that \(\sqrt{2}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. On proceeding with the proof, the student finds that both \(a\) and \(b\) are even. What decisive conclusion follows?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed, where \(p\) and \(q\) are coprime integers, which conclusion produces the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsA student says that \(\sqrt{3}\) is irrational because its decimal expansion does not terminate. Which step correctly turns this into a rigorous proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. What is the first conclusion obtained from \(3q^2=p^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsA student argues 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. Identify the student's error.Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsWhich initial assumption is made to prove the irrationality of \(\sqrt{2}\) by the method of contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsBefore proving the irrationality of \(\sqrt{3}\) by contradiction, in what form must \(\frac{p}{q}\) be assumed?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsWhy is the proof of \(\sqrt{2}\) valid without decimal expansion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsWhich of the following statements is correct in the contradiction proof that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsIn the proof that sqrt3 is irrational, if sqrt3 = p/q is assumed to be in lowest terms, why does a contradiction arise?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsWhich statement correctly describes the contradiction obtained while proving that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion produces a contradiction in the proof that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsWhat is the final contradiction in the proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium MathematicsA student assumes that \(\sqrt{2}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. From \(a^2=2b^2\), the student stops after concluding that \(a\) is even. Which conclusion is necessary to complete the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Medium