Question

In the group $(Z_5-\{\left[0\right]\},\cdot 5)$, the order of $\left[4\right]$ is

• A. $5$
• B. $3$
• C. $4$
• D. $2$

Answer 1

The correct option is D $2$

Consider the group $(Z_5-\{\left[0\right]\},.5)$

$Z_5-\left\{\right[0\left]\right\}=\left\{\right[1],[2],[3],[4\left]\right\}$

To find the order of each of the elements under ${.}_5\left[1\right]$ is the identity element under modular multiplication.

$\therefore O\left(\right[1\left]\right)=1$

$[2{]}^2=[4],[2{]}^3=\left[4\right]{.}_5\left[2\right]=\left[3\right],[2{]}^4=[3\left]{.}_5\right[2]=[1]$

$\therefore O\left(\right[2\left]\right)=4$

$[3{]}^2=[4],[3{]}^3=\left[4\right]{.}_5\left[3\right]=\left[2\right],[3{]}^4=[2\left]{.}_5\right[3]=[1]$

$\therefore O\left(\right[3\left]\right)=4$

$[4{]}^2=[1]$

$\therefore O\left(\right[4\left]\right)=1$

We have,

$O\left(\right[1\left]\right)=1,O\left(\right[2\left]\right)=4,O\left(\right[3\left]\right)=4,O\left(\right[4\left]\right)=2$

Hence, this is the answer.