Remainder of trinomial

Question

What will be the remainder if I divide $(a+b+c)^{333}-a^{333}-b^{333}-c^{333}$ by $(a+b+c)^3-a^3-b^3-c^3$. I have tried trinomial expention. But its still too big for long division. Is there a shorter method for this problem?

Answer 1

Since $$(a+b+c)^3-a^3-b^3-c^3=3(a+b)(a+c)(b+c)$$ and $$(a+b+c)^{333}-a^{333}-b^{333}-c^{333}=0$$ for $a=-b$, for $a=-c$, for $b=-c$

and $(a+b+c)^{333}-a^{333}-b^{333}-c^{333}$ is divided by $3$,

we obtain that the remainder is $0$.