Examine whether (8)ⁿ can end with the digit 5 for any n ϵ N.

If (8)ⁿ end with the digit 5, then the number should be divisible by 5.

This means the prime factorization of 8ⁿ should contain prime factor 5.

But (8)ⁿ = (2³)ⁿ and it does not have the prime factor 5 but have 2. So, the uniqueness of the Fundamental Theorem of Arithmetic guarantees that there are no other primes in the factorization of 8ⁿ.

, 5 is not present in the prime factorization, there is no natural number nor which 8ⁿ ends with digit 5.

So, 8ⁿ cannot end with digit 5.