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Math Help - Prove a congruence implication

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    Prove a congruence implication

    show that if 19a^{2} \cong b^{2} (mod\ 7) then 19a^{2} \cong b^{2} (mod\ 7^{2})
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  2. #2
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    Quote Originally Posted by mancillaj3 View Post
    show that if 19a^{2} \equiv b^{2} \!\!\!\pmod 7 then 19a^{2} \equiv b^{2} \!\!\!\pmod{7^{2}}
    First note that 19\equiv5\!\!\!\pmod7. So we want to know when a square can be congruent to five times a square (mod 7).

    The only squares (mod 7) are 0, 1, 2 and 4. And the only way for one of these to be congruent to five times another one is if they are both 0. Thus a and b must both be congruent to 0 (mod 7), and so a^2 and b^2 must both be congruent to 0 (mod 7^2).
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