Question

How many integers modulo${\mathbf{11}}^{\mathbf{3}}$ have inverses?$\mathbf{(}\mathbf{Note}\mathbf{:}{\mathbf{11}}^{\mathbf{3}}\mathbf{=}\mathbf{1331}\mathbf{)}$

Short answer

The number of integers coprime to${11}^3$ are 1210

Step-by-step solution

Explain Inverse modulo

The modular multiplicative inverse can be defined as the $\mathit{G}\mathit{C}\mathit{D}\mathbf{(}\mathit{a}\mathbf{,}\mathit{b}\mathbf{)}$must be equal to 1 and a, b are relatively prime.

Calculate the number of integers

Consider that the numbers$1,11,121,1331$ are the factors of ${11}^3$. It is given that ${11}^3=1331$, the number of integers can be calculated as follows,

$$\begin{gathered} {11}^3-1-\left({11}^2-1\right)=1331-1-\left(121-1\right) \\ =1330-\left(120\right) \\ =1210 \end{gathered}$$

Therefore, The number of integers coprime to ${11}^3$ are 1210