**Generalization of [this question](https://math.stackexchange.com/questions/4237066/remainder-theorem-technique)**

$$\frac{x^{2021}}{x^3 +x^2+x+1}$$
We wish to determine the remainder for the expression above. As in the linked question, write
$$x^{2021}=({x^3 +x^2+x+1})P(x)+ R(x)$$
To eliminate the term with P(x) let  

$$x^3 +x^2+x+1=0\Rightarrow x^4 =1$$
We must then have:
$$R(x)= x^{2021}=x^{2020}\times x=1 \times x =x$$

**Second Example**

The remainder of $x^{2023}$ when dividing by $x^3+x^2+x+1$ will be after reduction:
$$x^{2023}=x^3=-(x^2+x+1)$$
(You can write this more formally using the Remainder Theorem, but this is a shorter version).

**Further generalization**
We can generalize this technique. To find the remainder when dividing by
$$1+x+x^2+...+x^n$$
substitute
$$x^{n+1}=1,x\neq1$$

Is this correct?