Examine whether (7)ⁿ can end with the digit 5 for any n ϵ N.

If (7)ⁿ end with the digit 5, then the number should be divisible by 5.

This means the prime factorization of 7ⁿ should contain prime factor 5.

But (7)ⁿ does not have the prime factor 5. So, the uniqueness of the Fundamental Theorem of Arithmetic guarantees that there are no other primes in the factorization of 7ⁿ.

, 5 is not present in the prime factorization, there is no natural number nor which 7ⁿ ends with digit 5.

So, 7ⁿ cannot end with digit 5.