Prime factor of $17^3 - 5^3$

Question

I have the following question:

Find the largest prime factor of: $17^3 - 5^3$

Without any theory or tricks, I solved this the long way and I got $19$ as the answer. Though I would like to know if there any quick way of solving this. I understand the concepts of exponentiation and prime decomposition, but that doesn't help me here.

Answer 1

Use the identity $a^3-b^3=(a-b)(a^2+ab+b^2)$, where $a=17$ and $b=5$.

You will end up with $(12)(399)$.

Notice that $399=400-1=20^2-1^2$. Use the difference of two squares identity.

So $399=(20+1)(20-1)=21*19$.

Therefore the expression is equal to $12*21*19$. It is easy to see that the prime factors are $2, 3, 7$ and $19$.