Question

Find the remainder when the $41$-digit number $1234\ldots$ is divided by $8$. 

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Hint:

Divisibility by $8$ depends only on the last $3$ digits of the number.

Answer 1

The Correct Option is A

Interpreting $1234\ldots$ as the string $1234567891011\ldots$ continued until $41$ digits. Only the \emph{last three} digits matter mod $8$. Digits $1$–$9$ use $9$ places; remaining $32$ places are from two-digit numbers. That is $16$ numbers: $10$ to $25$. The final three digits are the last digit of $24$ and both digits of $25$, i.e. $425$. \[ 425 \div 8=53 \text{ remainder } 1. \] So the remainder is $1$.