Question

Find the remainder when  \[6^{\underbrace{66\cdots6}_{100 \text{ times}}}\] is divided by 10.
 

• A. 6
• B. 2
• C. 4
• D. 8

Hint:

Units digit cycles: $2,3,7,8$ have 4-cycles; $4,9$ have 2-cycles; $5$ and $6$ are \emph{fixed} (always 5 and 6).

Answer 1

The Correct Option is A

For any positive integer exponent $k\ge1$, the last digit of $6^k$ is always $6$. Therefore $6^{\text{(any positive integer)}}\equiv 6\pmod{10}$, regardless of how large the exponent is (here it's the 100-digit number consisting only of sixes). Hence the remainder upon division by $10$ is $\boxed{6}$.