<s>[INST] Show that $(a^2-b^2)(a^2-c^2)(b^2-c^2)$ is divisible by $12$ 
Let $a,b,c\in\Bbb N$ such that $a>b>c$. Then $K:=(a^2-b^2)(a^2-c^2)(b^2-c^2)$ is divisible by $12$.

My attempt : Since each $a,b,c$ are either even or odd, WLOG we may assume $a,b$ are both even or odd. For both cases, $a+b$ and $a-b$ are divisible by $2$ so $K$ is divisible by $4$. Note that any $n\in\Bbb N$ is one of $\overline{0},\overline{1},\overline{2}$ in $\operatorname{mod}3$. Well from this, I can argue anyway but I want to show $K$ is divisible by $3$ more easier or nicer way. Could you help?
 [/INST] To prove divisibility by $4$:
$(a^2-b^2)-(a^2-c^2)+(b^2-c^2)=0=\text{even}$
One of the addends must be even and this is possible with integers only if that is a multiple of $4$.
To prove divisibility by $3$:
Pigeonhole principle: at least two of $a^2,b^2,c^2$ must be multiples 9f $3$ or at least two must be one greater than a multiple of $3$. Either way the difference between the identified pair of squares is a multiple of $3$.
QED.
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