# Why is the sum modulo n of elements in $\mathbb{Z}_n^*$ equal to 0?

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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This is false when $n=2$. – Chris Eagle Mar 27 '12 at 11:47
Good point. Thanks for clarifying. – quantum Mar 27 '12 at 11:52
$ax+ny=1$ implies $(n-a)(-x) + n(y+x)=1$ so $(n-a,n)=1$ – Thomas Andrews Mar 27 '12 at 12:33

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

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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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Hint $\$ Negation $\rm\:n\to -n\:$ is an involution $\rm\:-(-n) \equiv n,$ so the cycles (orbits) of this permutation have length $2$ or $1.\$ But length $1$ is not possible since $\rm\: -a \equiv a\:$ $\Rightarrow$ $\rm\:n\:|\:2a\:$ $\Rightarrow$ $\rm\:n\:|\:2,\:$ by $\rm\:(n,a)=1.\$ Thus all cycles have length $2$ so have form $\rm\:(a,\:-a).\:$ These cycles partition the units into pairs which each sum to $0$, hence the entire sum of units is also $0$.

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

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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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