Hi! I have an issue with this math problem. Please help me!

Question

Prove that 47ⁿ - 1 is dividable by 8 for all positive and even whole numbers "n".

I know that you're supposed to use modulo to solve this issue, but i don't know how.

Answer 1

First, you can use Mathematical Induction.

Base case:

Let n = 2

47² - 1 = 2208 (which is divisible by 8)

Induction case:

Let (47^k - 1) mod 8 = 0 for all even positive integer k

Then (47^k+2 - 1) = (47²)(47^k) - 1

= 2209(47^k) - 1

= 2208(47^k) + (47^k - 1)

Since 2208 is divisible by 8, 2208(47^k) is divisible by 8

And (47^k - 1) is divisible by 8

Thus, in the conclusion, (47ⁿ - 1) is divisible by 8 for all even positive number