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Math Help - Number of solutions for x^2+x=0 mod n

  1. #1
    Senior Member vincisonfire's Avatar
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    Number of solutions for x^2+x=0 mod n

    (3) Consider the polynomial x^2 + x = 0 over Z/nZ.
    (a) Find an n such that the equation has at least 4 solutions.
    (b) Find an n such that the equation has at least 8 solutions.
    I found (but didn't proove) this :
    let a" be the number of solutions of x^2+x = 0 mod a and b" be the number of solutions of x^2+x = 0 mod b. Let Then if gcd(a,b)=1 number of solutions of x^2+x = 0 mod ab is a"*b".
    So far, for all values for which I tested it, it works. From that I can easily find the answer but I would like to know how to proove it.
    Thanks for your help!
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    Quote Originally Posted by vincisonfire View Post
    (3)let a" be the number of solutions of x^2+x = 0 mod a and b" be the number of solutions of x^2+x = 0 mod b. Let Then if gcd(a,b)=1 number of solutions of x^2+x = 0 mod ab is a"*b".
    I think that is just the chinese remainder theorem, isn't it? If gcd(a,b)=1 then for each solution of x^2+x = 0 mod a and each solution of x^2+x = 0 mod b there is a solution of x^2+x = 0 mod ab, and all solutions arise in that way.
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    Senior Member vincisonfire's Avatar
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    Reply

    Your right. The fact is we are gonna see this theorem in class in a month or so. I checked it out and found an easy solution for n such that there is 2^n solutions. Thank you though.
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