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Math Help - Proof help

  1. #1
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    Proof help

    For all integers a and b, if a^2 is congruent to b^2 (mod 20), then a is congruent to b (mod 5). prove true or false
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: Proof help

    Quote Originally Posted by lemondonut View Post
    For all integers a and b, if a^2 is congruent to b^2 (mod 20), then a is congruent to b (mod 5). prove true or false
    If If a^{2} \equiv b^{2}\ \text{mod}\ 20, then there is a constant k for which is...

    (a+b)\ (a-b)= 20\ k (1)

    From (1) is we derive that must be 5|(a+b) or 5|(a-b). If 5|(a-b) then is a \equiv b\ \text{mod}\ 5. If 5|(a+b) then for (1) it must be...

    (a-b)\ \frac{a+b}{5}= 4\ k \implies a-b= \frac{5}{a+b}\ 4\ k (2)

    ... so that 5|(a-b)...

    Kind regards

    \chi \sigma
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