Prove that $(a+b+c)^{333}-a^{333}-b^{333}-c^{333}$ is divisible by $(a+b+c)^{3}-a^{3}-b^{3}-c^{3}$

Question

Prove that $$(a+b+c)^{333}-a^{333}-b^{333}-c^{333}$$ is divisible by $$(a+b+c)^{3}-a^{3}-b^{3}-c^{3},$$ where $a,,b,c -$ integers, such that $(a+b+c)^{3}-a^{3}-b^{3}-c^{3}\not =0$

My work so far:

$(a+b+c)^{3}-a^{3}-b^{3}-c^{3}=3(a+b)(b+c)(c+a)$

Let $333=3\cdot111=3t,$ where $t=111$.

$(a+b+c)^{333}-a^{333}-b^{333}-c^{333}=(a+b+c)^{3t}-a^{3t}-b^{3t}-c^{3t}$.

I need help here.

Answer 1

Hint $1:$ Let $f(a)=(a+b+c)^{333}- a^{333} -b^{333} -c^{333}.$

$f(-b)=(-b+b+c)^{333} -(-b)^{333} -b^{333} -c^{333}=0$

So, by Factor theorem we get $(a+b)$ as a factor of the given expression.

Similarly, $(b+c)$ and $(c+a)$ are factors of the given expression.

Hint $2:$ Now, to prove the divisiblity of the expression by $3$, use Fermat's little theorem.

$a^2 \equiv 1 \pmod 3 \Rightarrow a^{333} \equiv a\pmod 3$

Similarly,

$b^{333} \equiv b \pmod 3$
$c^{333} \equiv c \pmod 3$
$(a+b+c)^{333} \equiv (a+b+c)\pmod 3$

So, the given expression is $\equiv 0 \pmod 3$

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As the given expression is divisible by each of these four factors, it is divisible by $(a+b+c)^3 -a^3-b^3-c^3$.