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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।

MathematicsIn the contradiction proof that √3 is irrational, \(p^2=3q^2\) gives \(3\mid p^2\). Which fact is used to conclude that \(3\mid p\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsWhile proving the irrationality of \(\sqrt{3}\) by contradiction, what conclusion about \(p\) is drawn from \(p^2=3q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsIf \(\sqrt{3}=\frac{p}{q}\) is assumed in lowest terms and \(p^2=3q^2\) is obtained, what is the valid basis for concluding that \(3\mid p\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsWhat is the correct argument involving the prime number 3 in the proof of √3?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 MathematicsSuppose \(\sqrt{3}=\frac{p}{q}\) is written in lowest terms. If \(3\mid p^2\), which of the following conclusion is valid?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 17Hard MathematicsA student assumes \(\sqrt{3}=\frac{a}{b}\), where \(a,b\) are coprime positive integers. From \(a^2=3b^2\), the student concludes that 3 divides \(a\). Which rule justifies this conclusion?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsIf (p^2=3q^2) and (p=3k), what combined conclusion about (p) and (q) follows from (q^2=3k^2)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhich statement is correct in the proof of the irrationality of \(\sqrt{3}\), based on \(a^2=3b^2\), where \(a\) and \(b\) are coprime integers?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Hard MathematicsWhich option shows the correct cause-effect relation used in the proof of \(\sqrt{3}\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsWhich of the following statements is essential for proving the irrationality of \(\sqrt{3}\) by the contradiction method?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsIn the proof that \(\sqrt{3}\) is irrational, suppose \(\sqrt{3}=p/q\) is in lowest terms and squaring gives \(p^2=3q^2\). What correctly fixes a student's error that \(3\mid p^2\) does not imply \(3\mid p\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsWhile proving \(\sqrt{3}\) irrational by contradiction, suppose \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers. What follows immediately from \(p^2=3q^2\)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsSuppose a student writes \(\sqrt{3}=\frac{p}{q}\), where \(p\) and \(q\) are coprime, while proving that \(\sqrt{3}\) is irrational. From \(p^2=3q^2\), which conclusion about \(p\) is correct?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsWhich statement is used to move from p² to p in the proof of √3?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 18Medium MathematicsIn the proof that \(\sqrt{3}\) is irrational, \(a^2=3b^2\) gives \(3\mid a^2\). Which rule justifies the next step?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsIf (p^2=3q^2) and (p=3k), why will (q) be divisible by (3)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 16Medium MathematicsFrom (q^2=3r^2) what conclusion is obtained about (q)?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20Easy MathematicsIn the proof that \(\sqrt{3}\) is irrational, if \(p^2\) is divisible by 3, which conclusion about \(p\) is correct?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 20Easy MathematicsWhich number-theoretic fact is used decisively while proving the irrationality of \(\sqrt{3}\) by contradiction?Class 9Number SystemsProof of irrationality of square root 2 and square root 3Level 19Easy