Problem on Mathematical Induction

[Karen:)]

Q: Prove that 8^(n) − 3^(n) is divisible by 5 for all natural numbers n.

Hint:
- P(1): 8^(1) - 3^(1) = 5, which is divisible by 5. Therefore, P(1) is true.
- Assume P(n) is true for some natural numbers n, i.e. 8^(n) - 3^(n) = 5M for some integer M

How can I solve this question? Many thanks!

 

[Harry_the_cat]

Assume that 8ⁿ-3ⁿ = 5*M.

You now need to prove that
8ⁿ⁺¹ - 3ⁿ⁺¹ is a multiple of 5

8ⁿ⁺¹ - 3ⁿ⁺¹
=8*8ⁿ - 3*3ⁿ
=8*8ⁿ - 8*3ⁿ +5*3ⁿ
=8*(8ⁿ - 3ⁿ) + 5 (3ⁿ)
=.... can you finish it?