An exercise regarding polynomials I guess it is a simple exercise though I'm not very good at polynomials. It asks:
Find $m,n,p,q$ natural numbers such that the polynomial $X^m+X^n+X^p+X^q$ is divisible by $x^3+x^2+x+1$.
Thank you in advance.

Hint : show that $X^m+X^n+X^p+X^q$ is divisible by $X^3+X^2+X+1$ iff $m,n,p,q$ form a complete set of residues modulo $4$. Note that the value of a power $X^j$ modulo $X^3+X^2+X+1$ only depends on the value of $j$ modulo $4$.
In other words, the most general solution to your problem is $X^{4q_1}+X^{4q_2+1}+X^{4q_3+2}+X^{4q_4+3}$ where $q_1,q_2,q_3,q_4$ are nonnegative integers.
