Question

Prove that ${25}^n-{20}^n-8^n+3^n$ is divisible by $5$.

Answer 1

Consider the given expression, $8^n-3^n$.

We to prove $8^n-3^n$ is divisible by $5.$

We know that,

$[a^n-b^n=(a-b)(a^{n-1}+a^{n-2}.b+a^{n-3}{b}^2+a^{n-4}{b}^3+............+{b^n}^{-1})]$

Put $a=8,b=3,$we get

$8^n-3^n=(8-3)(8^{n-1}+8^{n-2}.3+8^{n-3}{3}^2+8^{n-4}{3}^3+............+{3^n}^{-1})$

$8^n-3^n=5(8^{n-1}+8^{n-2}.3+8^{n-3}{3}^2+8^{n-4}{3}^3+............+{3^n}^{-1})$

Which is divisible by $5.$hence proved.