Question

Find the inverse of 5 under multiplication modulo 11 on ${{Z}_{11}}$ .

Answer 1

To solve this question what we will do is, firstly we will find out the multiplicative identity for modulo 11. And then using the definition of multiplicative inverse for modulo 11, we will find the inverse of 5 from the set of ${{Z}_{11}}$.

Complete step-by-step answer:
Before we solve this question, we need to see what are multiplicative identity elements and multiplicative inverse elements of any number in a finite set S.
Now, let we have set S and a be any element belonging to set S.
So, the identity element is denoted by ‘e’ and is defined as such an element if we have $a,e\in S$ then for any operation o, we have $a\circ e=e\circ a=a$ for all $a,e\in S$.
And, inverse element of element a, where $a,e\in S$for any operation o, we have $a\circ {{a}^{-1}}={{a}^{-1}}\circ a=e$ for all $a,e\in S$.
Now, we have set ${{Z}_{11}}$ and ${{Z}_{11}}$ is set containing the first 11 positive integers from 0.
So, ${{Z}_{11}}=\\{0,1,2,3,4,5,6,7,8,9,10,11\\}$
For finding the inverse element, we need to find the identity element first.
Multiplication modulo is denoted by ${{\times }_{11}}$ and we have $a,e\in S$then, it is defined as
$a{{\times }_{11}}e=a$
$ae=a$
Or, e = a
So, for multiplication modulo 11, we have identity equals to 1.
And for multiplicative inverse for modulo 11, $a{{\times }_{11}}x=1$, where $x={{(a)}^{-1}}$
Here, in question we are asked to find an inverse of 5 for multiplication modulo 11.
Now, let x be inverse of 5, then $x={{(5)}^{-1}}$
Using definition of inverse of element for multiplication modulo 11, we get
$\left( 5.{{(5)}^{-1}} \right)=1$
Or, $\left( 5x-1 \right)=0$
So, we can say that $\left( 5x-1 \right)$must be multiple of 11.
Now, from set ${{Z}_{11}}=\\{0,1,2,3,4,5,6,7,8,9,10,11\\}$, we can see that for x = 9
$\left( 5(9)-1 \right)=44$ and which is multiple of 11.
So, the inverse of 5 under multiplication modulo 11 on ${{Z}_{11}}$ is 9.

Note: To solve this question one must know the meaning of inverse of element and identity of element on any operation say, o. one must know what does ${{Z}_{11}}$ actually means which is set of first positive integers including 0. Try to avoid calculation mistakes.