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If p is the prime positive …

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If p is the prime positive integer . Prove that √p is an irrational number

    • Posted by Nandini Verma 4 years, 7 months ago

    CBSE > Class 10 > Mathematics
    • 1 answers

    Sia ? 4 years, 3 months ago

    For any prime positive integer p, √p is an irrational number.

    Let us assume that √p is a rational number.

    Then, there exist positive co-primes a and b such that :-

    √p = a/b

    p = a²/b²

    b²p = a²

    p divide a²

    p divides a.

    a = pc ( positive integer c. )

    Now, b²p = a²

    b²p = p²c²

    b² = pc²

    p divide b²

    p divides b

    Therefore, p/a and p/b

    This contradicts the fact that a and b are co-primes.

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