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 $(5-1)(5+1)(5+2)$

Hint: for induction step
$$ 5^{3(n+1)}+2\cdot 5^{2(n+1)} -5^{n+1}-2 $$
$$= 5^{3n}\cdot(124+1)+2\cdot 5^{2n}\cdot(24+1) -5^{n}\cdot(4+1)-2$$ 
$$= (5^{3n}+2\cdot 5^{2n} -5^{n}-2) +   (5^{3n}\cdot 124+2\cdot 5^{2n}\cdot 24 -5^{n}\cdot 4) $$
$$= (5^{3n}+2\cdot 5^{2n} -5^{n}-2) +   4\cdot (5^{3n}\cdot 31+2\cdot 5^{2n}\cdot 6 -5^{n}). $$
