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Prime Divisibility

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

MathematicsWhich property of the prime number 3 is used decisively in the proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which statement correctly justifies the conclusion \(3\mid p\) from \(3\mid p^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIn proving the irrationality of ext{\(\sqrt{3}\)} , Ravi assumes that ext{\(\sqrt{3}=p/q\)} , where ext{\(p\)} and ext{\(q\)} are coprime. From ext{\(p^2=3q^2\)} , he says that only ext{\(p\)} is divisible by 3 and nothing can be concluded about ext{\(q\)} . Which fact corrects his error?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 about \(p\) and \(q\) contradicts this assumption?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Expert MathematicsIf \(p/q\) is in lowest terms and \(p^2=3q^2\), which conclusion establishes the contradiction in the proof that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed in lowest terms and \(p^2=3q^2\) is obtained, which conclusion establishes the contradiction in the proof of irrationality?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, if \(\frac{p}{q}\) is in lowest terms, which immediate conclusion follows from \(3\mid p^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsRima assumes that \(\sqrt{3}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Which conclusion in her proof establishes a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIf \(n\) is an integer and \(3\mid n^2\), which conclusion used in the proof of the irrationality of \(\sqrt{3}\) is certainly true?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 65Expert MathematicsIf someone writes (p=3q) from (p^2=3q^2), what is the correct correction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsIn the contradiction proof that \(\sqrt{3}\) is irrational, assume \(\sqrt{3}=\frac{p}{q}\), where \(\gcd(p,q)=1\). This gives \(p^2=3q^2\). Which principle is correctly applied from \(3\mid p^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsAssume that a/b is in lowest terms and obtain a² = 3b². Which conclusion about a is essential in the proof that √3 is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Expert MathematicsWhich divisibility rule is crucial in the proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsIn the proof by contradiction for the irrationality of \(\sqrt{2}\) and \(\sqrt{3}\), which prime-number property for an integer \(n\) is used decisively?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsWhich property is used decisively in the proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsA student assumes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, and obtains \(p^2=3q^2\). Which next step is logically valid for reaching a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Expert MathematicsWhich prime-divisibility property is crucial in a proof by contradiction that \(\sqrt{3}\) is irrational?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. Which conclusion follows from this assumption and produces a contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIn a proof that \(\sqrt{3}\) is irrational, if \(\frac{p}{q}\) is assumed to be in lowest terms and \(p^2=3q^2\) is obtained, which conclusion establishes the contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Hard MathematicsIf 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 18Hard