Question

If $n>2$, prove that $n-1$ is an element of order 2 in role="math" localid="1652333239524" $U_n$.

Short answer

We proved that $n-1$is an element of order 2 in $U_n$, if $n>2$.

Step-by-step solution

n-1 in powers of 2

Suppose $U_n$ is a group of units which is a group of integers under multiplication modulo n.

Let $n>2$.

Let $n-1\in U_n$

We have to show that $n-1$is an element of order 2.

Now, ${(n-1)}^2=n^2-2n+1=n(n+1)+1$

To find the order of n-1

From step 1,

We can write,

${(n-1)}^2=1$(mod n) in $U_n$.

This implies that the order of $n-1$is 2 in $U_n$.

This proves that $n-1$ is an element of order 2 in $U_n$, if $n>2$.