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

$$5^{3n} + 2 \cdot5^{2n} - 5^{n} - 2=5^n\cdot5^{2n}+2 \cdot5^{2n} - 5^{n} - 2=$$
$$5^{2n}( 5^{n} +2)-( 5^{n}+2)=( 5^{n} + 2)(5^{2n} - 1)=$$
$$=( 5^{n} + 2)(5^{n} + 1)(5^{n} -1)$$
