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I'm talking about the set $$\mathbb{Z}_n^* = \{x \in \mathbb{Z}_n : \text{gcd}(x,n)=1\}$$ I noticed that for $n>2$, if you add all the elements in the set, you get $0\mod{n}$. Can someone explain why this is? I also noticed that if $a$ is in the set, then so is $n-a$, but how can I go about showing that $\text{gcd}(n-a,n)=1$?

For example, $$\mathbb{Z}_8^* = \{1,3,5,7\}$$ And $1+7 = 8$, $3+5=8$. So $1+3+5+7 \equiv_8 0$

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  • $\begingroup$ This is false when $n=2$. $\endgroup$ Mar 27, 2012 at 11:47
  • $\begingroup$ Good point. Thanks for clarifying. $\endgroup$
    – quantum
    Mar 27, 2012 at 11:52
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    $\begingroup$ $ax+ny=1$ implies $(n-a)(-x) + n(y+x)=1$ so $(n-a,n)=1$ $\endgroup$ Mar 27, 2012 at 12:33

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Your remark is what you need.

Assume that $a \in \mathbb{Z}_n^*$ and some integer $k > 1$ divide both $n-a$ and $n$. Then $k$ would divide their difference $n - (n-a) = a$. Hence $k$ divide $n$ and $a$ and hence $a$ cannot belong to the set. Thus $n-a \in \mathbb{Z}_n^*$

Now you just need to observe that $a + (n-a) = n = 0$ in $\mathbb{Z}_n^*$ and $$ \sum_{a \in \mathbb{Z}_n^*} a = \sum_{a \in \mathbb{Z}_n^*, a < n/2} (a + n-a) = 0. $$

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This is an edit of my previous answer.

The following proof is incomplete and more complex than the simple argument that uses the simmetry $a\to-a$ of $\Bbb Z_n^\times$ employed, but is possibly the simplest case of a useful argument.

Assume first that $n=p$ is a prime. Let $x\in\Bbb Z_p^\times$. The multiplication by $x$ (i.e. the map $a\mapsto ax$) defines a permutation of $\Bbb Z_p$ which restricts to a permutation of $\Bbb Z_p^\times$. Therefore $$ \sum_{a\in\Bbb Z_p^\times}a= \sum_{a\in\Bbb Z_p^\times}(xa)= x\sum_{a\in\Bbb Z_p^\times}a. $$ If $n\neq2$ we can always choose $x\neq1$ and the above identity shows that $\sum_{a\in\Bbb Z_p^\times}a$ must be $0$.

If $n$ is not prime the problem is that a congruence $$ xA\equiv A\bmod n\qquad\qquad(*) $$ does not imply $A\equiv 0\bmod n$ even when $x\neq1$ is invertible in $\Bbb Z_n$ because $A$ may not be cancelled. The latter situation happens when $A\notin\Bbb Z_n^\times$, i.e. ${\rm gcd}(A,n)>1$.

So, assume $A\not\equiv0\bmod n$ and write $d={\rm gcd}(A,n)$, $A=Bd$ and $n=dm$ and note that ${\rm gcd}(B,m)=1$. The congruence $(*)$ becomes $$ xB\equiv B\bmod m $$ which now implies $x\equiv1\bmod m$ because $B\in\Bbb Z_m^\times$. But the latter conclusion contradicts the possibility to choose $1\neq x\in\Bbb Z_m^\times$ when $m\neq2$, which is guaranteed by the fact that the canonical map $$ \Bbb Z_n^\times\longrightarrow\Bbb Z_m^\times $$ is surjective.

In particular this proves the result when $n$ is odd and leaves open the only possibility that $$ \sum_{a\in\Bbb Z_{2m}^\times}a=m $$ in $\Bbb Z_{2m}$.

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    $\begingroup$ Beware that this proof only works for prime $\,n\,$ since it implicitly assumes nonzero elements are cancellable i.e. cancel $S\neq 0\,$ in $\,xS = S\Rightarrow x = 1\Rightarrow\!\Leftarrow,\,$ so $\,S=0,\,$ where $S$ is the sum. Please edit to restrict (vs. delete) the answer since it is being used as a dupe target (in the prime case). $\endgroup$ Dec 6, 2021 at 10:13
  • $\begingroup$ @BillDubuque, hi Bill, funny thing to have to edit a post about 10 years after it was written. I modified the argument so to make it work for all $n$ odd but couldn't find a truely elementary way (except the same $a\to -a$ business, of course) to deal with the last case. $\endgroup$ Dec 9, 2021 at 0:42
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If $d$ divides both $n-a$ and $n$ then it divides their difference, $n-(n-a)$, which is $a$.

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Every common divisor of $n-a$ and $n$ is also a common divisor of $a$ and $n$. This shows that if $a$ is in $(\mathbb Z/n\mathbb Z)^\times$ then also $n-a$ is in this set. The pair $(a,n-a)$ sums up to $n$ which is congruent to $0$ modulo $n$. Also, the numbers $a$ and $n-a$ are distinct since $a \equiv n-a \pmod n$ would imply $n|2a$, hence $n|2$ (because $n$ and $a$ are coprime). For $n\neq 2$ this is a contradiction. For $n=2$, however, the statement is actually false since $(\mathbb Z/2\mathbb Z)^\times = \{1\}$ which does not sum up to $0$ modulo $2$.

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Coprimes to $\,n\,$ are closed under negation, so $\,a\,$ and $-a\,$ $\rm\color{#c00}{pair}$ up and cancel out of the sum, so the sum is $\equiv 0\,$ (they always pair, else $\, a \equiv -a\,$ so $\,n\,|\,2a,\ (n,a)\!=\!1$ $\Rightarrow$ $\,n\,|\,2,\,$ contra $\,n\!>\!2).\,$ $\bf\small QED$


Said $\rm\color{#c00}{pairing}$ can be viewed group theoretically: negation $\,n\to -n\,$ is an involution $\,-(-n) \equiv n$, i.e. it's a permutation of order $2$ so its cycles (orbits) have length $2$ or $1$. These cycles partition $\Bbb Z_n^*$ into pairs $(a,-a),\ a\not\equiv -a,\,$ and singles $(a),\ a\equiv -a$. By above there can be no singles, and each pair sums to $\,0\,$ hence so too does the entire sum.

This is a prototypical example of Wilson's theorem for groups.

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  • $\begingroup$ Coprimes are closed under negation since coprimes are precisely the invertibles (units) $\bmod n$, and this is clear for invertibles: $aa^{-1}\equiv 1\Rightarrow (\color{#0a0}{-a})(\color{#c00}{-a^{-1}})\equiv 1,\,$ so $(\color{#0a0}{-a})^{-1}\equiv \color{#c00}{-a^{-1}},\,$ i.e. $\,1/(-a) = -1/a.\,$ We use the rep $\,n-a\equiv -a\,$ in the standard nonnegative residue system $\,\{0,1,2,\ldots, n-1\}.\ \ $ $\endgroup$ May 8, 2023 at 1:44
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Observe that if $\gcd(n-a,n)=d$ then $d|n$ and $d|(n-a)$. Since $d|n$, we have $d|a$. Hence $d|\gcd(n,a)=1$. So you have $d=1$.

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