Question

If $\mathit{n}\mathbf{ }\mathbf{>}\mathbf{ }\mathbf{2}$, prove that $\mathit{n}\mathbf{ }\mathbf{-}\mathbf{ }\mathbf{1}$is an element of order 2 in ${\mathit{U}}_{\mathbf{n}}$.

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 modulolocalid="1659452132605" $n$.

Let $n>2$.

Let $n-1\in U_n$

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

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

Step 2: To find order of  n-1

From step 1,

We can write,

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

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

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