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Prove that the following number is irrational: √6

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This question is from real numbers in which we have given a number √6 and it is already asked in various examinations and we have to show that the given number is a irrational number.

Kindly give me a detailed solution of this question

RS Aggarwal, Class 10, chapter 1D, question no 3(i)


1 Answer

  1. let us suppose that 6 is rational number.

    There exist two co-prime numbers , say p and q
    So √6=p/q

    Squaring both sides , we get

    Which shows that , is divisible by 6
    this implies , p is divisible by 6
    Let p=6a for some integer a

    Equation (1) implies = 6=36
     is also divisible by 6
    q is divisible by 6

    6 is common factors of p and q
    but this contradicts the fact that p and q have no common factor.
    our assumption is wrong thus √6 is irrational

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