$f(n)=7^{6n} - 6^{6n}$, where $n$ is a positive integer. $f(n)=7^{6n} - 6^{6n}$, where $n$ is a positive integer. find the divisors of $f(n)$ for odd and even values of $n$. Is there a general solution for the divisors.
$$f(1)=7^6-6^6=(7^3)^{2}-(6^3)^{2}$$
$$f(1)=(7-6)(7^2+(7)(6)+6^2)(7^3+6^3)$$
$$f(1)=(1)(127)(7^3+6^3)$$ 

As you have said, you always have $7^{6n}-6^{6n}=(7^{3n})^2-(6^{3n})^2=(7^{3n}-6^{3n})(7^{3n}+6^{3n})$
As Mark Bennet has said $7^{3n}+6^{3n}=(7^n+6^n)(7^{2n}-7^n6^n+6^{2n})$
Also, $7^{3n}-6^{3n}=(7^n-6^n)(7^{2n}+7^n6^n+6^{2n})$
So we have  $7^{6n}-6^{6n}=(7^n-6^n)(7^{2n}+7^n6^n+6^{2n})(7^n+6^n)(7^{2n}-7^n6^n+6^{2n})$
Alpha confirms this is the best we can do for $n=1$, but there are more for $2$ through $5$.
