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Math Help - Explanation

  1. #1
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    Explanation

    Can someone explain me why n^2+n+1 isn't dividable with 125?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by DenMac21
    Can someone explain me why n^2+n+1 isn't dividable with 125?
    Consider n^2+n+1 modulo 10, for n=10k+r with
    r\in \{0,1,2,3,4,5,6,7,8,9\}. You will find n^2+n+1
    is never divisible by 5 and so cannot be divisible by 125.

    RonL
    Last edited by CaptainBlack; February 7th 2006 at 02:49 PM.
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  3. #3
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    Quote Originally Posted by DenMac21
    Can someone explain me why n^2+n+1 isn't dividable with 125?
    If,
    n^2+n+1\equiv 0\mod 5
    Noticing that 5 is a prime, look at its discrimanant ( b^2-4ac).

    The above congruence has a solution if and only if its discrimant is a quadradic residue.
    Since b^2-4ac=-3\equiv 2.
    Apply Euler's Criterion to determine if something is a quadradic residue we get that,
    2^{\frac{5-1}{2}}\not \equiv 1\mod 5
    Thus, n^2+n+1 is never divisible by 5, thus how can it be divisible by 125=5^3?
    Thus, we have shown it is impossible.
    Q.E.D.

    To remind you Euler's Criterion states, a is a quadradic residue of p if and only if,
    a^{\frac{p-1}{2}}\equiv 1\mod p
    Last edited by ThePerfectHacker; February 7th 2006 at 03:34 PM.
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