Alternating last digits

Let $A$ be a single digit positive integer such that

• the last digit of $A, A^3, A^5, A^7$ are all the same, and it is equal to $B$ .
• the last digit of $A^2 ,A^4, A^6, A^8$ are all the same, and it is equal to $C$ .

If $B$ and $C$ are distinct digits. What is $B+C$ ?