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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।

MathematicsA student assumes that \(\sqrt{3}\) is rational and writes it as \(\frac{p}{q}\) in lowest terms, where \(p\) and \(q\) are coprime. Which conclusion correctly follows from \(p^2=3q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student assumes that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. During the proof, the student obtains \(p^2=3q^2\). What is the correct conclusion needed to establish a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student claims that if \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then \(p^2=2q^2\) implies only that \(p\) is even; nothing can be concluded about \(q\). What is the error in the student's reasoning?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsWhich assumption is required at the beginning of a proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Which conclusion from \(p^2=3q^2\) is necessary to reach a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsWhy must the fraction be taken in lowest terms when assuming \(\sqrt{2}=\frac{p}{q}\) in the proof that \(\sqrt{2}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsIf \(p\) and \(q\) are coprime integers and \(p^2=3q^2\), which conclusion is essential in the proof that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsSuppose \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers. Which of the following properties is crucial for proving that this assumption is contradictory?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsAarav assumes that \(\sqrt{3}=\frac{m}{n}\), where \(m\) and \(n\) are coprime. After obtaining \(m^2=3n^2\), he says that divisibility of \(m^2\) by 3 does not prove that \(m\) is divisible by 3. Which fact corrects Aarav’s error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsWhile starting a proof by contradiction for the irrationality of \(\sqrt{2}\), Riya assumes that \(\sqrt{2}=p/q\), where \(p\) and \(q\) are integers. Which additional condition on \(p\) and \(q\) is necessary to make the proof valid?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsA student assumes that \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\ne0\), and concludes from \(p^2=2q^2\) that both \(p\) and \(q\) are even. If the student did not assume \(p\) and \(q\) to be coprime, why is this conclusion not automatically a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIn the proof by contradiction that \(\sqrt{2}\) is irrational, assume \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion contradicts this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIn the proof by contradiction for the irrationality of \(\sqrt{3}\), assume \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which conclusion from \(p^2=3q^2\) is essential for the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed, where p and q are integers, why must \(\frac{p}{q}\) be taken in lowest terms in the proof by contradiction of its irrationality?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsWhen proving the irrationality of \(\sqrt{3}\) by the contradiction method, which assumption is made at the start?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIn a proof by contradiction that \(\sqrt{3}\) is irrational, it is assumed to be \(\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which conclusion contradicts the initial assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIn a proof by contradiction, assume that \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion from \(p^2=3q^2\) leads to the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsSuppose \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Which contradiction follows from this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsWhile proving \(\sqrt{3}\) irrational by contradiction, we assume \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. Which conclusion at the end of the proof establishes the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsWhat is the main purpose of assuming \(\sqrt{2}=\frac{p}{q}\) in lowest terms in the standard proof by contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert