Prove $5 \mid (3^{4n} - 1)$ by induction I need to prove by induction that $5 \mid 3^{4n} - 1$ for $n \ge 1$. Base case is true, so now I have to prove that $5 \mid 3^{4(n+1)} - 1$.
I did 
$$= 3^{4n+4} -1$$
$$= 3^{4n} 3^{4}-1$$
I guess I need to make a $3^{4n}-1$ appear somewhere to use the inductive hypothesis, but I don't know how. Should I use logarithm? 

Without induction:
$$3^{4n}-1=81^n-1^n=(81-1)\left(81^{n-1}+81^{n-2}+\cdots+1\right)$$
