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Math Help - prove congruence

  1. #1
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    Question prove congruence

    Prove that if a is an odd integer, then a^2 is congruent to 1(mod8)
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  2. #2
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    Quote Originally Posted by mandy123 View Post
    Prove that if a is an odd integer, then a^2 is congruent to 1(mod8)
    a=8k\pm 1,8k\pm 3.
    Now square them.
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  3. #3
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    Question

    so when I square them how exactly is that going to help me prove that a is congruent to 1(mod8)?

    Am I just suppose to give examples
    like if k=3
    then it would be
    625 which is congruent to 1(mod8) and so on, or do I need to continue the proof?
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  4. #4
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    Quote Originally Posted by mandy123 View Post
    so when I square them how exactly is that going to help me prove that a is congruent to 1(mod8)?
    (8k\pm 1)^2 = 8(8k^2) + 8(\pm 2k) + 1 = 8(8k^2 \pm 2k)+1 thus it is congruent to 1 mod 8.

    (8k\pm 3)^2 = 8(8k^2) + 8(\pm 6k) + 9 = 8(8k^2 \pm 6k + 1)+1 thus it is congruent to 1 mod 8.
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  5. #5
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    oh, ok, i see now how you did it. I was just squaring them completely differently than you. Thank you so much!
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