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

8-151.

5^{1} − 1 = 4

Assume that 5* ^{k}* − 1 is divisible by 4, therefore 5

*− 1 = 4*

^{k}*A*.

5^{k}^{ + 1} − 1 =

5(5)* ^{k}* − 1 =

5(5)

*− 5 + 4 =*

^{k}5(5

*− 1) + 4 =*

^{k}20

*A*+ 4 = 4(5

*A*+ 1)

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.