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Which number-theoretic fact is used decisively while proving the irrationality of \(\sqrt{3}\) by contradiction?विरोधाभास विधि से \(\sqrt{3}\) की अपरिमेयता सिद्ध करते समय कौन-सा संख्या-सिद्धांत तथ्य निर्णायक रूप से प्रयोग किया जाता है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: If \(3\mid p^2\), then \(3\mid p\)यदि \(3\mid p^2\), तो \(3\mid p\)
Let \(\sqrt{3}=p/q\) in lowest terms. From \(p^2=3q^2\), \(3\mid p^2\) implies \(3\mid p\), since 3 is prime. Substituting \(p=3k\) also gives \(3\mid q\), a contradiction. Exam tip: always state that \(p,q\) are coprime.मान लें \(\sqrt{3}=p/q\), जहाँ \(p,q\) सहाभाज्य हैं। \(p^2=3q^2\) से \(3\mid p^2\), इसलिए 3 अभाज्य होने के कारण \(3\mid p\)। फिर \(p=3k\) रखने पर \(3\mid q\) भी मिलता है, जो विरोधाभास है। \(9\mid p\) आवश्यक नहीं है।
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