Coprime elements

**slevvio** · January 14th 2011, 03:33 AM

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.

**tonio** · January 14th 2011, 03:47 AM

Originally Posted by slevvio

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

**slevvio** · January 14th 2011, 04:00 AM

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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