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Math Help - proving two statements to be true

  1. #1
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    proving two statements to be true

    I have two questions, and any help would be greatly appreciated.

    1. Show that if a = b (mod m) and c = d (mod m), (assume a,b,c,d, and m are all integers with m>=2), then a-c = b-d (mod m).

    I started this off by saying that if a=b (mod m), then (a-b)/m = s, where s is some integer, and similarly, that if c=d (mod m), then (d-c)/m = t, where t is some integer. Actually, I don't think this helps any =/.


    2. Prove that if n is an odd positive integer, then n^2 = 1(mod 8).

    Well, I plugged in odd positive integers for n, and yeah, it's true. But how do I go about proving it?

    Thanks guys.
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  2. #2
    Super Member Matt Westwood's Avatar
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    Quote Originally Posted by Walcott89 View Post
    I have two questions, and any help would be greatly appreciated.

    1. Show that if a = b (mod m) and c = d (mod m), (assume a,b,c,d, and m are all integers with m>=2), then a-c = b-d (mod m).

    I started this off by saying that if a=b (mod m), then (a-b)/m = s, where s is some integer, and similarly, that if c=d (mod m), then (d-c)/m = t, where t is some integer. Actually, I don't think this helps any =/.


    2. Prove that if n is an odd positive integer, then n^2 = 1(mod 8).

    Well, I plugged in odd positive integers for n, and yeah, it's true. But how do I go about proving it?

    Thanks guys.
    1. Express a = b(mod m) as a = rm + b, and c=d(mod m) as c = sm+d, subtract one from the other and show that a-b = km + (c-d) where k is some integer.

    2. Express n = 2k + 1, square it and see what factors it.
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