<s>[INST] 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 $(5-1)(5+1)(5+2)$
 [/INST] Basis(n=1):  $5^3+50-5-2=168 = 4*42$
Inductive step: assume that $5^{3n}+2*5^{2n}-5^n+1 = 4k, k\in \mathbb{Z}$. Then,
$$
5*(5^{3n})+10*{5^{2n}}-5*5^n + 1 = 5(5^{3n}+2*5^{2n}-5^n)+1 = 5(4k-1)+1=20k-4=4(5k-1).
$$
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