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Math Help - Prove or disprove the following statement; modulo, variables, integers, etc.

  1. #1
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    Prove or disprove the following statement; modulo, variables, integers, etc.

    I am working through a past exam as part of my study for uni, and I've come across a question that I can't work out.


    Prove or disprove the following statement:
    For all intergers n, 3(n^2)-5 is either congruent to 2 modulo 4 or congruent to 3 modulo 4.


    I really don't even know where to start with this question, I have rearranged the two modulo terms into 2mod4=4k+2 (for some integer k) and 3mod4=4p+3 (for some integer p), but have no idea whether this is of any help in solving the question or not.


    Any help would be greatly appreciated,
    Thanks.
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    Re: Prove or disprove the following statement; modulo, variables, integers, etc.

    Quote Originally Posted by lucaswilson94 View Post
    I am working through a past exam as part of my study for uni, and I've come across a question that I can't work out.


    Prove or disprove the following statement:
    For all intergers n, 3(n^2)-5 is either congruent to 2 modulo 4 or congruent to 3 modulo 4.


    I really don't even know where to start with this question, I have rearranged the two modulo terms into 2mod4=4k+2 (for some integer k) and 3mod4=4p+3 (for some integer p), but have no idea whether this is of any help in solving the question or not.


    Any help would be greatly appreciated,
    Thanks.
    Hint: What possible values can n^2 take a modulo 4?

    -Dan
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  3. #3
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    Re: Prove or disprove the following statement; modulo, variables, integers, etc.

    If n is even, that is, if n= 2k for some integer k, then 3n^2- 5= 3(4k^2)- 5.
    If n is odd, that is, if n= 2k+ 1 for some integer k, then 3n^2- 5= 3(2k+ 1)^2- 5= 3(4k^2+ 4k+ 1)- 5.

    Factor "4"s out of those.
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  4. #4
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    Re: Prove or disprove the following statement; modulo, variables, integers, etc.

    Hi,
    Here are some facts about congruence that apply to your problem. They also apply to many other problems involving congruence.

    Prove or disprove the following statement; modulo, variables, integers, etc.-mhfdiscretemath5.png
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