### Home > PC > Chapter 8 > Lesson 8.2.6 > Problem8-151

8-151.

Use mathematical induction to prove that $5^{n} − 1$ is divisible by 4.

$5^{1} − 1 = 4$

Assume that $5^{k} − 1$ is divisible by $4$, therefore $5^{k} − 1 = 4A$.

$5^{k+ 1} − 1 =$
$5\left(5\right)^{k} − 1 =$
$5\left(5\right)^{k} − 5 + 4 =$
$5\left(5^{k} − 1\right) + 4 =$
$20A + 4 = 4\left(5A + 1\right)$
So $5^{k + 1} − 1$ is a multiple of $4$.

Hence, by mathematical induction we have proven that $5^{n} − 1$ is divisible by $4$ for all natural numbers.