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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{3}\\) is written as \\(a/b\\), where \\(a\\) and \\(b\\) are coprime integers, which conclusion produces the contradiction in the proof of its irrationality?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18HardMathematicsIn a proof that \(\sqrt{3}\) is irrational, a student assumes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, and obtains \(p^2=3q^2\). The student directly writes that \(q\) is divisible by 3. Which statement is needed to make the reasoning valid?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18HardMathematicsIf ext{\(\sqrt{3}\)} is assumed to be ext{\(p/q\)} , where ext{\(p\)} and ext{\(q\)} are coprime integers, which fact produces the contradiction in the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18HardMathematicsWhich of the following statements is an essential part of the proof by contradiction that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsIn a proof by contradiction that \(\sqrt{2}\) is irrational, suppose \(\sqrt{2}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Which final condition contradicts this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsWhich skipped step would make the proof of \(\sqrt{2}\) incomplete?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsIn the standard proof, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which number-theoretic fact is needed to obtain a contradiction from \(p^2=3q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsSuppose \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. On squaring, we get \(a^2=3b^2\). Which conclusion is justified at this stage?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed with \(p\) and \(q\) coprime, and the proof shows that both \(p\) and \(q\) are divisible by 3, what does this establish?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsIf \(\sqrt{3}=p/q\) is assumed with \(p\) and \(q\) coprime, which condition contradicts this assumption in the proof of irrationality?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsSuppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion produces the contradiction in the proof that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsIn a proof by contradiction, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. If \(3\mid p^2\) is obtained, which conclusion must follow next?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsIf \(p\) and \(q\) are coprime integers and \(p^2=3q^2\), which conclusion necessarily follows?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsSuppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. After obtaining \(p^2=2q^2\), which conclusion correctly advances the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsIf \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, is assumed, which conclusion creates a contradiction in the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsIn a proof by contradiction that \(\sqrt{2}\) is irrational, why must \(p\) and \(q\) be chosen coprime when assuming \(\sqrt{2}=\frac{p}{q}\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsIf the square of a rational number is 3, what conclusion is obtained about its numerator and denominator in the proof that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsRiya assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. After concluding from \(p^2=3q^2\) that \(p\) is divisible by 3, which is the correct next statement to obtain a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17HardMathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, assume \(\sqrt{3}=p/q\), where \(p\) and \(q\) are coprime. Which conclusion produces a contradiction to this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16HardMathematicsIn the proof of \(\sqrt{2}\), if a student ends the proof after only writing \(a\) is even, what is the error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard