Proving by induction $5^{3n} + 2 \cdot 5^{2n} - 5^{n} - 2$ is divisible by $4$ I want to prove the following twice. Once by induction then again by any other method.
$$5^{3n} + 2 \cdot 5^{2n} - 5^{n} - 2$$ is a multiple of 4 for all nonnegative integers n.
Let n=0 , since it is the first nonnegative integer
$$5^{3(0)}+2*5^{2(0)}-5^{0}-2 = 0 $$

Factoring gives us $(5-1)(5+1)(5+2)$

$$5^{3n} + 2 \cdot 5^{2n} - 5^{n} - 2=(5^n+2)(5^{2n}-1)=(5^n+2)(5^n-1)(5^n+1)$$
Do you see middle term is divisible by $4$?
