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Math Help - Coprime elements

  1. #1
    Senior Member slevvio's Avatar
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    Coprime elements

    Let \mathbb{Z}_{p^r} = \{ 0,1, \ldots, p^r -1\} be the ring of integers modulo p^r, a power of a prime. I was wondering if someone could help explain why there are p^{r-1} elements that are coprime to p^r in this ring? I am getting tangled up in knots and would appreciate any help with this.

    Thanks very much.
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  2. #2
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    Quote Originally Posted by slevvio View Post
    Let \mathbb{Z}_{p^r} = \{ 0,1, \ldots, p^r -1\} be the ring of integers modulo p^r, a power of a prime. I was wondering if someone could help explain why there are p^{r-1} elements that are coprime to p^r in this ring? I am getting tangled up in knots and would appreciate any help with this.

    Thanks very much.
    In every string of length p of the form kp, kp+1,\ldots,kp+(p-1) there are exactly p-1 numbers coprime with p^r, and there are p^{r-1} such strings between 0\,\,and\,\,p^r, thus the number is
    actually p^{r-1}(p-1) and not p^r...

    T onio
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  3. #3
    Senior Member slevvio's Avatar
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    ah sorry i meant, NOT coprime. Thanks very much for your answer, there was a mistake in my notes which was causing confusion but I can see from what you said
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