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

1. 6: Any positive-integer power of a number ending in \( 6 \) also ends in \( 6 \), since \( 6\times6=36 \) always regenerates the same last digit, regardless of how large the exponent is.
2. 2, 4, 8: None of these can occur here, because the last digit of powers of \( 6 \) never changes, unlike bases such as \( 2,3,7,8 \) that cycle through four different last digits.

The last digit of \( 6 \) raised to any positive power is always \( 6 \), so the remainder on division by \( 10 \) is \( 6 \).