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Divisibility

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

MathematicsIf p is not divisible by 3 in p² = 3q², what contradiction appears?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsA student assumes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, and obtains \(p^2=3q^2\). The student shows only that \(3\mid p\) and declares a contradiction. Which step correctly completes the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsIf a student stops after proving only that a is divisible by 3 in the proof of √3, what is the main error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIn a proof by contradiction for the irrationality of \(\sqrt{3}\), a student obtains \(p^2=3q^2\). Given that \(p\) is divisible by 3, which next step correctly advances the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\sqrt{3}=\frac{p}{q}\) is assumed in lowest terms and \(p^2=3q^2\) is obtained, which conclusion is necessary to reach the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsWhat is the basis of 3 ∣ a² ⇒ 3 ∣ a in the proof of √3?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsA student claims that if the square of an integer is divisible by 3, then the integer itself is divisible by 3. What is the correct evaluation of this claim?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIn the proof of (\sqrt{3}), if (a) is not divisible by (3), what inconsistency arises from (a^2=3b^2)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIn 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 18Hard MathematicsIf the square of an integer is divisible by 3, which conclusion about the integer must be true?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms, which conclusion about \(p\) and \(q\) produces the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIn 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 17Hard MathematicsIf assuming \(\sqrt{3}=\frac{p}{q}\) in lowest terms leads to \(p^2=3q^2\), which conclusion proves a contradiction in this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsIn the proof of √3, what kind of step is it to write directly that v is divisible by 3 from u² = 3v²?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsIf (u^2=3v^2) and (u=3t), what is the combined conclusion about (u) and (v)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsIn the proof of (\sqrt{3}), if (u) is not divisible by (3), what conflict occurs from (u^2=3v^2)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\), a student assumes \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime. The student obtains \(a^2=3b^2\). Which conclusion is correct for proceeding with the proof?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsLet \(p\) and \(q\) be coprime positive integers. In a proof by contradiction for the irrationality of \(\sqrt{3}\), if \(3\mid p^2\), which conclusion is necessary?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, suppose \(\sqrt{3}=\frac{p}{q}\) is in lowest terms. What conclusion about \(p\) follows from \(p^2=3q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard 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