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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।

MathematicsWhich of the following expressions is divisible by \,a-b\, for all values of the variables?Class 9Exploring Algebraic IdentitiesAlgebraic identitiesLevel 63Expert MathematicsUnder which condition is the sum of the first n natural numbers a multiple of n?Class 9Sequences and ProgressionsSum of first n natural numbersLevel 61Expert MathematicsFor which type of positive integer \(n\) is the sum of the first \(n\) natural numbers exactly divisible by \(n\)?Class 9Sequences and ProgressionsSum of first n natural numbersLevel 61Expert MathematicsFor which positive integers \(n\) is the sum \(S_n\) of the first \(n\) natural numbers exactly divisible by \(n\)?Class 9Sequences and ProgressionsSum of first n natural numbersLevel 60Hard MathematicsWhich sequence is a sequence of even numbers?Class 9Sequences and ProgressionsSequencesLevel 45Easy MathematicsIn the proof by contradiction that \(\sqrt{3}\) is irrational, which property is used to conclude \(3\mid p\) from \(3\mid p^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn the proof of √3, 3 | h² implies 3 | h. Which broader principle does this illustrate?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard 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 MathematicsWhich of the following statements is essential in a proof by contradiction that \(\sqrt{3}\) is irrational?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 MathematicsA student claims that if \(3p^2=q^2\), then \(p\) must be divisible by 3. What is the correct evaluation of the claim?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 MathematicsIf (h) is not divisible by (3) and (h^2=3k^2), what inconsistency appears?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed to be in lowest terms and \(p^2=3q^2\) is obtained, which conclusion establishes the contradiction in this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsWhich of the following statements is correctly used in the proof that sqrt(3) is irrational in number systems?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert 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 says that if \(\sqrt{3}=\frac{p}{q}\), then \(p^2=3q^2\) only implies that \(p\) is divisible by 3; nothing can be concluded about \(q\). What is the student's error?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIf (m) is not divisible by (3) and (m^2=3n^2), what inconsistency is obtained?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsIn the proof by contradiction that \(\sqrt{3}\) is irrational, suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. From \(p^2=3q^2\), which conclusion is necessary?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIn a proof by contradiction for the irrationality of \(\sqrt{3}\), which conclusion is necessary to proceed after obtaining \(a^2=3b^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert