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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।

MathematicsReema says that if \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then \(p^2=2q^2\) only shows that \(p\) is even; therefore \(\sqrt{2}\) is not proved irrational. Which essential point is missing from Reema's argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn the standard proof by contradiction, suppose that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. From \(p^2=3q^2\), which deduction is essential for obtaining the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsWhile proving the irrationality of \(\sqrt{2}\) by contradiction, if \(\sqrt{2}=\frac{p}{q}\) where \(p\) and \(q\) are coprime, which conclusion necessarily follows from \(p^2=2q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. On squaring, the student gets \(p^2=3q^2\) and concludes that \(p\) is divisible by 3. What is the next essential step to complete the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsWhy is it necessary to take the fraction \(\frac{p}{q}\) in lowest terms in the contradiction proof that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student claims that \(\sqrt{3}\) is rational and writes it as \(\frac{p}{q}\) in lowest terms, where \(p,q\) are coprime. If \(p^2=3q^2\), what is the error in this claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring, \(p^2=2q^2\) is obtained. Which conclusion follows correctly?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn a proof by contradiction, suppose \(\sqrt{2}=\frac{h}{k}\), where \(h\) and \(k\) are coprime integers. Which conclusion contradicts this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. After obtaining \(3q^2=p^2\), which of the following conclusion is correct?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn a proof by contradiction that \(\sqrt{3}\) is irrational, assume \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which conclusion from \(p^2=3q^2\) is needed to establish the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsSuppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion follows from this assumption and contradicts the fraction being in lowest terms?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student assumes \(\sqrt{2}=p/q\), where \(p\) and \(q\) are coprime integers, to prove that \(\sqrt{2}\) is irrational. From \(p^2=2q^2\), the student writes \(p=2m\) and obtains \(q^2=2m^2\). The student says that since \(p\) and \(q\) are coprime, \(q\) must be odd. What is the correct correction to this statement?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn the contradiction proof of the irrationality of \(\sqrt{3}\), what does writing \(\frac{p}{q}\) in lowest terms mean?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn the contradiction proof of the irrationality of \(\sqrt{3}\), we assume \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime. If both \(p\) and \(q\) are finally found to be divisible by 3, which conclusion is correct?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn the proof by contradiction for the irrationality of sqrt(3), before assuming sqrt(3) = p/q, which condition is essential for the fraction p/q?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIf ext{\(\sqrt{3}\)} is assumed to be ext{\(\frac{p}{q}\)}, where ext{\(p\)} and ext{\(q\)} are coprime, which conclusion produces the contradiction in a proof by contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. On squaring, the student gets \(p^2=2q^2\). Which of the following is the valid next inference?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student claims that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. From \(p^2=3q^2\), which conclusion about \(p\) is necessary?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student assumes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, to prove that \(\sqrt{3}\) is irrational. After obtaining \(3q^2=p^2\), the student says, “\(p\) is divisible by 3, so a contradiction has been reached.” Which statement correctly identifies the gap in the argument?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsSuppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion creates the contradiction in the proof by contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert