If r is rational and r^2 is an integer, prove that r is an integer....Math Proofs- irrational and integers?

If r is rational and r^2 is an integer, prove that r is an integer.

please help me figure this out

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  • 8 years ago
    Best Answer

    r is rational, so r = a / b for some integers a and b, where b is not zero and GCD(a,b) = 1.

    r^2 = a^2 / b^2 is an integer.

    Thus, a^2 = kb^2 for some integer k.

    GCD(a,b) =1 implies GCD(a^2 , b^2) = 1

    We have: GCD(a^2 , b^2) = GCD(kb^2 , b^2) = b^2 = 1

    This implies b = 1 or = -1

    Since r = a / b, it is the case that r = a / 1 = a or r = a / -1 = -a

    Either way, since a is an integer, it follows that r = a and r = -a are also integers.

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