Is $512^3+675^3+720^3$ a composite number? Q & A style
A question for High school students (Calculators not allowed !):
Is $512^3+675^3+720^3$ a composite number?
I am posting this in Q & A style. Any suggestion will be appreciated.

Well there may be so many ways to answer. One of my teachers in a math-camp gave the following way out. I think its a good one. Any other better ways of solution will be appreciated.
Let $a=512, b=675, c=720$. Now the number looks like $a^3+b^3+c^3$ but we require a sort of $3abc$ term to resolve it into factors. First we factorize $a,b,c$ into prime factors. So, $a=2^9, b=3^3\times 5^2, c=2^4\times 3^2\times 5$. Now it can be seen that $2c^2=3ab$. Hence, $a^3+b^3+c^3=a^3+b^3-c^3+2c^2c$ and thus the problem can be solved. It can then be seen that the given number is composite.  
